## August 2021

## Special Focus: Valves, Pumps and Turbomachinery

# PSV sizing: An alternative solution to the homogeneous direct-integration method

Pressure safety valves (PSVs) are designed to protect personnel and plant properties from an overpressure occurrence in equipment or piping by relieving fluid to a safe location.

Pressure safety valves (PSVs) are designed to protect personnel and plant properties from an overpressure occurrence in equipment or piping by relieving fluid to a safe location. While other limitations exist, such as material selection and temperature-pressure rating, a proper sizing can guarantee and optimize protection. The first (and potentially most) important step in the design of a safety valve is to size it as accurately as possible.

Various approaches—theoretical and empirical— to size such a device for all fluids have been derived, developed, published and reviewed by many authors.^{1–9} The theoretical equations presented are based on the homogeneous equilibrium model (HEM), assuming thermal and mechanical equilibrium for the fluid passing through the valve nozzle at any cross-section perpendicular to the flow direction (i.e., thermo-physical properties, including density, are uniform at normal direction).

For single-phase fluids, sizing equations are well established and known. They can be solved analytically or numerically and are provided in many references. These methods require thermo-physical properties of the fluid at the stagnation state (relief condition). However, the approaches for two-phase relief sizing are more complex and require the consideration of many factors.^{2–9} The homogenous nonequilibrium model, for instance, may be considered for delayed flashing after the liquid reaches the saturation pressure in short nozzles. Also, due to flashing, the gas/vapor phase expands, resulting in higher gas-phase velocity compared to liquid-phase velocity. The velocity difference (known as slip effect) is incorporated into the two-phase models by defining slip ratio/ratio of the gas velocity to the liquid velocity at the throat of the nozzle. To take such effects into account, some guidelines have been provided for two-phase fluids and liquids flashing inside the nozzle.^{2,3,4,9}

Among all developed methods for PSV sizing, the homogeneous direct-integration method (HDIM) is the most reliable approach in industrial applications and recommended by the American Petroleum Institute.^{1} The method is general, simple and accurate enough to be applied for any fluid in the process of PSV sizing; it can be used as long as the fluid density as a function of pressure is available and can be precisely estimated by thermodynamic databases. Also, it is not subject to the many assumptions or restrictions applied to other methods.^{2,3} The HDIM is also based on the homogeneous equilibrium model described above. Applying general steady-state volumetric energy balance to a homogeneous fluid flowing through an adiabatic-reversible (isentropic) process within an ideal nozzle results in a theoretical equation (Eq. 1), which relates mass flux (*G*) at the nozzle throat to the fluid density (*ρ*) and the pressure (*P*) variations:

(1)

The subscripts *t* and *r* represent the properties at the nozzle throat and relief conditions, respectively. The derived equation can be used to size a PSV for any homogeneous fluid. The relationship between the fluid density and pressure through an isentropic process must be known.^{1,3} The theoretical mass flux is then corrected using a dimensionless constant known as discharge coefficient (*K _{d}*), which relates the theoretical ideal nozzle mass flux to the actual mass flux and published experimentally by relief valve vendors.

Having the mass flowrate (*m˙*) to be relieved, the required nozzle throat cross section area (*A*) can be calculated using Eq. 2:^{1}

(2)

where ∏ *K _{i}* is the product of all applicable correction coefficients, including discharge coefficient, backpressure (

*K*), viscosity (

_{b}*K*) and combination (

_{v}*K*) correction factors. The throat pressure varies between relief pressure (

_{c}*P*) and backpressure (

_{r}*P*) in Eq 3:

_{b}*P _{r}* <

*P*≤

_{t}*P*(3)

_{b}Once the relief valve opens, the fluid begins flowing though the nozzle, resulting in increasing fluid velocity and decreasing fluid pressure in the flow direction. The maximum velocity will be achieved at the throat (minimal cross-section area). If the velocity approaches the sound velocity at a throat pressure less than the total backpressure, the flow becomes choked at the throat providing the maximum mass flux (e.g., for gases, two-phase fluids and vapors). Otherwise, the maximum mass flux is obtained once the throat pressure approaches the total backpressure (e.g., for liquids). The objective of the PSV sizing method is to find the maximum mass flux at the throat pressure by solving Eq. 1.

A numerical method—trapezoidal rule or Simpson’s Rules—is commonly performed to solve Eq. 1 and then find the throat pressure and the maximum mass flux.

The accuracy of this approach depends strongly on the selected step size, as well as the numerical integration method. The truncation errors of the numerical integration should be minimized to achieve an acceptable solution. This can be accomplished by reducing the step size and by using a higher-order integration rule. A smaller step size decreases the truncation error, resulting in a more accurate solution; however, more calculation steps (numerous isentropic phase-equilibrium calculations) are needed, so it is a time-consuming procedure. Using the higher-order integration rule brings more complexities to the solution.

A few analytical solutions have been presented for an individual fluid to overcome the numerical issues.^{1,6,8} Based on Eq. 1 and a pressure-specific volume correlation, for instance, a universal mass flux equation (similar to the Omega method for critical flow) and an equivalent critical pressure equation were applied to predict mass flux for any fluid at any condition except flashing flow of initially subcooled liquid.^{6} The pressure-specific volume correlation had two parameters that can be calculated using one, two or three data points. Although the models showed a good accuracy for the mass flux and the critical pressure estimations, they were limited to the fluid state and condition.

To avoid such issues related to numerical integration and to present a universal model applicable to any fluid at any condition, an alternative solution (analytical) is proposed to solve Eq. 1. The main objectives of the presented work are to bring all PSV sizing equations under one umbrella by using Eq. 1, to reduce the number of isentropic process calculations to a few data points, to solve Eq. 1 analytically by targeting the valve throat pressure, and to calculate mass flux at the throat for sizing a PSV. This requires that the density of the fluid with respect to pressure variations can be expressed as a linear, a second-order polynomial, or an exponential function with high accuracy. Additionally, for each fluid state, a detailed example is provided to show the applicability of the approach for any fluid and to provide guidelines for PSV sizing.

## Theory: An analytical approach

The approach provided is easy to follow:

- Determine the fluid state and properties, including pressure, temperature and density at the relief condition (Point 1).
- Determine the total backpressure and the pressure range (
*P*–_{r}*P*)._{b} - Based on the authors’ experiences, 3–6 data points (including the relief condition) are enough to recognize the trend of fluid density with respect to pressure changes. Divide the pressure range into approximately
*n*equal increments (Eq. 4):

(*P*–_{r}*P*)/_{b}*n*(4)

where*n*would be 2, 3, 4 and 5, based on the number of selected data points between 3 and 6. In this article, three data points are selected to examine the proposed model—in general, using more data points provides a better trend line for the fluid density-pressure relationship, leading to a more accurate model. - Decrease the relief pressure by one increment through an isentropic process and find the fluid density at the new state. In a proprietary simulator
^{a}, for example, use an expander model palette with 100% efficiency to simulate an isentropic process and ensure that the mass entropy remains constant for both initial and final states. Repeat this step two times until the pressure approaches the total backpressure. - Plot the three data points (density vs. pressure) and find a proper trend for density as a function of pressure. Our experiences show that the data can be fitted to a linear, a second-order polynomial or an exponential function.
- The regression process can be calculated by many math software tools (in Excel, for example, use a combination of
*index*and*linest*functions). Fit the data points to one of the above functions and find all required constants, as well as*R*^{2}. Although other statistical rules exist,*R*^{2}is sufficient to judge the validity of the curve fit. Estimate the throat nozzle pressure from one of the following equations, calculate corresponding mass flux, and then size the PSV using Eq. 2. - In all cases 0 ≤
*P*<_{t}*P*. If_{r}*P*≤_{b}*P*<_{t}*P*, then use_{r}*P*as the throat pressure. Otherwise, total backpressure should be used as the throat pressure._{t }

## A linear function

For liquids, the density as a function of pressure usually can be expressed linearly (Eq. 5):

*ρ* = *aP* + *b* (5)

where *a* and *b* are the equation constants. Substituting Eq. 5 into Eq. 1 and taking the integral of the function results in Eq. 6:

(6)

To maximize the mass flux at the throat (Eq. 7):

(7)

Applying Eq. 7 to Eq. 6 results in an explicit equation, which relates the throat pressure to the relief pressure as (Eq. 8):

(8)

Eq. 6 gives the maximum mass flux, *G _{max}*

^{2}, at this pressure.

## A second-order polynomial

If the fluid density-pressure trend can be fitted to a second-order polynomial, then (Eq. 9):

*ρ* = *aP*^{2} + *bP* + *c* (9)

Substituting Eq. 9 into Eq. 1, taking the integration results in three cases (depends on *a, b* and *c* coefficient values and signs) to calculate the mass flux, and then applying Eq. 7 to the obtained mass flux, the throat pressure can be targeted.

## Case 1: 4*ac* – *b*^{2} < 0

(Eqs. 10, 11 and 12):

(10)

where

(11), (12)

In this case, the throat pressure must satisfy the following implicit equation (Eq. 13):

(13)

## Case 2: 4*ac* – *b*^{2} > 0

(Eqs. 14, 15 and 16):

(14)

where

(15), (16)

The throat pressure must satisfy the following implicit equation (17):

(17)

## Case 3: 4*ac* – *b*^{2} = 0

(Eqs. 18 and 19):

(18)

and

(19)

where *ac* > 0 and then √(*c*/*a*) exists.

Eqs. 13 and 17 are implicit functions of the throat pressure and should be solved by a proper algorithm. In Excel or MATLAB, one can use a “*solver*” function or “*fzero*” command to find the throat pressure, respectively. Both methods need an initial guess, which should be between the relief pressure and the total backpressure. Note that the upper and lower limits are the relief pressure and zero, respectively. Eq. 19, however, is an explicit function of the throat pressure but a less likely case, based on the authors’ experience in PSV sizing.

## An exponential function

When a liquid (subcooled or saturated) is flashed, its density with respect to pressure variations may be fitted with an exponential function with a higher *R*^{2}. In such a case, Eq. 20 can be used:

ρ = *ae ^{bP}* (20)

Following the same procedure, Eqs. 21 and 22 can be obtained:

(21)

where the throat pressure for the maximum mass flux calculations can be evaluated by Eq. 22:

(22)

Constant density. Subcooled liquid density is a weak function of pressure and can be generally assumed constant within the pressure range *P _{b}* ≤

*P*<

*P*. In such a condition, Eq. 1 can be simplified to Eq. 23:

_{r}(23)

Note that the flow is not choked for this condition, and the downstream total backpressure is used as the throat pressure.

Examples have been provided to support the proposed method. For all examples, three data points—at the relief pressure, a pressure in the middle of the pressure range, and the total backpressure—are selected to express a proper trend for density-pressure relationship.

# RESULTS

## Example A: Two-phase fluid

A balanced PSV was designed to protect a horizontal separator if the liquid outlet flow is blocked (governing case for an actual design). The drum design pressure was 1,000 kPag (PSV set pressure). The inlet fluid was a mixture of an oil (mostly heavy oil), water and hydrocarbon vapor with flowrate of 61,552 kg/hr (582.9 kg/hr vapor and 60,969.1 kg/hr liquid) at 427.5 *K*, which must be relieved to a storage tank (no credit was taken for the gas outlet). The entire process was simulated using a proprietary process simulator^{a} and fluid properties; the hydraulics of vessels and pipes were also estimated. At relief conditions—427.5 K and 1,201.1 *kPaa* (10% overpressure and atmospheric pressure 101.1 *kPaa*)—the properties were calculated (**TABLE 1**) and the two-phase flow discharge coefficient was estimated to be *K _{d}* = 0.873 using the method presented by Leung.

^{10}Additionally, the total backpressure was calculated to be 231

*kPag*(332.1

*kPaa*) at the rated flowrate, resulting in

*K*= 1 for the balanced PSV (Crosby backpressure curves were used). The viscosity correction factor was estimated to be

_{b}*K*= 0.990.

_{v}Through an isentropic process, the fluid relief pressure was reduced two times (with a 345-*kPa* interval) and the density was calculated (**TABLE 2**). The data was fitted to a linear as well as a second-order polynomial using a proper regression method (**FIG. 1**). The throat pressure and the maximum mass flux were then calculated. The calculation details and results are summarized in **TABLE 2**.

The valve is sized by HDIM as well as a two-point Omega method.^{8} For HDIM, the density of the fluid was calculated with a 7-*kPa* interval pressure reduction; the mass flux and the required area were then calculated from the numerical solution of Eq. 1. For the Omega method, however, the required properties at 90% of the relief (saturation) pressure (1,081 *kPaa*) were estimated through an isentropic process (**TABLE 1**) and the valve was then sized. The maximum mass fluxes and required nozzle areas calculated by both methods are found in **TABLE 2** for comparison.

For two-phase flow PSV sizing, Simpson^{9} presented a method based on a two-point interpolation scheme for properties estimation and numerical integration of Eq.1. The first point was at the relief pressure and the second one was at a pressure between 50% to 95% of the relief pressure (in this example, 700 *kPaa*). Along an isentropic path, the fluid was flashed from 1,201.1 *kPaa* to 700 *kPaa*; the input data are summarized in **TABLE 1**. Using provided two-point data, the density was estimated and then the mass flux and the required area were calculated from the numerical solution of Eq. 1 with a pressure interval of 10 *kPa*. The results are summarized in **TABLE 2** for comparison.

Although all methods led to a calculated API PSV size of “*L*”, the two-point Omega method resulted in a less conservative nozzle size, which can be related to the linear interpolation data behind the model. Based on the authors’ PSV sizing experiences, once the density-pressure trend deviates from a linear behavior, the two-point Omega method is not recommended for two-phase relief PSV sizing, as it may result in an undersized PSV. However, a linear trend will result in the same PSV size using all methods, including the Omega method, which is consistent with the linear-interpolation assumption applied to the two points of the Omega method (see the two-phase example in supporting examples).

For two-phase flow relief, the proposed analytical method is generally accurate enough to size PSVs; however, if slip effect or HNE model^{3} is a concern, the proposed approach is still valid for PSV sizing. In such cases, the density as a function of pressure is estimated, corrected using the slip or the HNE model, and then applied for the illustrated procedure to size PSV for two-phase relief.

## Example B: Subcooled or saturated liquid flashing at the throat

The most challenging part of the proposed method is when the PSV sizing process is dealing with saturated or highly subcooled liquid flashing (the liquid reaches the saturation pressure flowing through the nozzle). In this condition, two different regimes have different density trends. In the subcooled regime, the density is almost constant, whereas high-density variations can be observed after the liquid starts flashing. These variations cannot be fitted by an individual equation (**FIG. 2**). Fortunately, a solution exists for such a condition. The integral has an interval addition property allowing us to divide it into two intervals—in our case, one interval for subcooled liquid and another one for flashed liquid—that hold the same conditions. Therefore, Eq. 1 can be rearranged into (Eq. 24):

(24)

The maximum mass flux of the subcooled regime, *G _{s}*

^{2}, can be estimated using Eq. 23 as well, if the density does not vary significantly over the pressure range. The flashing regime can be fitted into a second-order polynomial or an exponential function. To show the capability of the proposed solution under such a condition, consider the example C.2.3.2 of API-520-1

^{1}, in which 378.5 L/min (11,588 kg/hr) of subcooled propane should be relieved at 2,073.2

*kPaa*and 288.7

*K*. The downstream total backpressure was 170.2

*kPaa*, leading to a backpressure correction factor unit. The discharge coefficient was 0.65 and the viscosity correction factor was assumed to be 1. The valve was sized using a two-point Omega method, resulting in the maximum mass flux and required area of 36,890 kg/(s – m

^{2}) and 134.5 mm

^{2}, respectively.

To size the PSV using the proposed method, the subcooled propane at the relief condition was flashed using an isentropic path (through an expander with 100% efficiency in the simulator^{a} environment). At the first stage, the pressure was decreased to find the boiling point (*P _{sat}* = 722.2

*kPaa*). Two sets of data (a subcooled set between

*P*and

_{r}*P*and a flashed set between

_{sat}*P*and

_{sat}*P*) were then generated and are presented in

_{b}**TABLE 2**. The subcooled set was fitted to a second-order polynomial, whereas the flashed set was fitted by two functions, a seconded-order polynomial and an exponential function. At the final stage, the maximum mass fluxes were added and the required area was calculated. The results are also summarized in

**TABLE 2**. Both polynomial and exponential functions show almost the same results.

Eq. 23 can be used to evaluate the subcooled mass flux using an average density of 509.28 kg/m^{3} to be *G _{s}*

^{2}= 1,370,914,531 kg

^{2}/(s

^{2}– m

^{4}), showing the same result.

Note: If the relief pressure is at or very close to (slightly subcooled liquid) the saturation pressure, then it is recommended that the subcooled regime be ignored and the PSV be sized in one step using Eq. 1. For such a condition, the flashed data starting from the saturation point are fitted by an exponential function, but the relief pressure should be used for Eqs. 21 and 22. Simpson^{9} provided an example to size a rupture disk for relieving 181,437 kg/hr of 703.3 *kPaa* condensate saturated at 689.5 *kPaa*. The discharge coefficient and the total backpressure were assumed to be 0.62 and atmospheric pressure, respectively. Using the two-point model prediction and the numerical solution of Eq. 1, the throat pressure, the maximum mass flux and the required area were calculated to be 688.8 *kPaa*, 5,136.3 kg/(s^{2} – m^{4}) and 15,806 mm^{2}, respectively. Three data points at 689.5, 668.8 and 655.0 *kPaa* were selected and fitted to an exponential function, leading to Eq. 25:

*ρ* = 1.4692 × 10^{–4}*e*^{0.02265P} R^{2} = 0.9928 (25)

Using the relief pressure of 703.3 *kPaa*, the throat pressure, the maximum mass flux, and the required area can be calculated using Eqs. 22, 21 and 2, respectively, as:

*P _{t}* = 672.7

*kPaa*

*G*= 5,181.3 kg/(s – m

^{2})

*A*= 15,690 mm

^{2}

## Takeaway

To avoid numerous calculation steps of the numerical solution of Eq. 1 to size pressure relief valves, an analytical approach is derived to solve an integral part of the homogeneous direct-integration method by fitting three density-pressure data points to a linear, a second-order polynomial or an exponential equation (more data points, up to six, are recommended to evaluate density-pressure trend and relationship). The approach is straightforward, targeting the throat pressure, calculating the mass flux, and then sizing the PSV.

For all possible fluid states to be relieved, one example was provided and compared with other sizing methods to support the approach. The results show that a second-order polynomial can represent a density-pressure relationship for all fluids through an isentropic pressure reduction process. However, the equations derived for targeting the throat pressure are implicit and must be solved by an appropriate algorithm. The linear model provides an explicit equation with less accuracy for calculating the throat pressure.

In the case of subcooled liquid flashing through the nozzle, an exponential function may offer a more accurate fitting curve to the density-pressure relationship, leading to an explicit expression for the throat pressure calculation as well. In general, the presented analytical solution is accurate enough to be applied to all fluid states for PSV sizing. It can be extended to take into account the homogenous non-equilibrium directed integration model, as well as the slip model recommended for two-phase flow relief.^{2,3,9} The density at each step should be corrected accordingly before using the analytical solution. Also, for two-phase relief sizing, if the density-pressure relationship shows a high degree of non-linearity, the two-point Omega method is not recommended for PSV sizing due to the linear interpolation assumption behind the model development. **HP**

**NOTES**

^{a} Aspen HYSYS

**ACKNOWLEDGEMENT**

The authors would like to show their gratitude and appreciation to Tony Steinicke (WSP Process Manager) and Tao Song (WSP Lead Process Engineer) for their support and feedback.

**NOMENCLATURE**

*a, b* and *c *Coefficients of fitted equations*A *Nozzle cross-section area, mm^{2}*C _{P}* Liquid specific heat capacity, kJ/kg-K

*G*Mass flux, kg/sec-m

^{2}

*h*Specific enthalpy, kJ/kg

*K*Backpressure correction factor, dimensionless

_{b}*K*Combination correction factor, dimensionless

_{c}*K*Discharge coefficient, dimensionless

_{d}*K*Viscosity correction factor, dimensionless

_{v}*m˙*Mass flowrate, kg/hr

*P*Absolute pressure,

*kPaa*

*R*

^{2}A statistical measure of goodness of regression, dimensionless

*T*Temperature, K

*v*Specific volume, m

^{3}/kg

**Greek letters**

*α* Actual gas volume fraction*ρ* Density, kg/m^{3}

**Subscripts**

*b *Properties at total backpressure*C* Calculated*f* Flashed*lg* Difference between gas and liquid property*l* Liquid*max * Maximum*r* Properties at relief pressure*sat* Saturation*s * Subcooled*S* Selected*t* Properties at throat pressure*v* Vapor

**LITERATURE CITED**

- American Petroleum Institute (API), “Sizing, selection and installation of pressure-relieving devices—Part 1: Sizing and selecting,” 9th Ed., 2014.
- Darby, R., “Evaluation of two-phase flow models for flashing flow in nozzles,”
*Process Safety Progress,*2000. - Darby, R., “Size safety-relief valves for any conditions,”
*Chemical Engineering,*2005. - Darby, R. et al., “Select the best model for two-phase relief sizing,”
*Chemical Engineering Progress,*2001. - Darby, R. et al., “Properly size pressure-relief valves for two-phase flow,”
*Chemical Engineering,*2002. - Kim, J. S. et al., “Sizing calculations for pressure-relief valves,”
*Chemical Engineering,*2013. - Leung, J. C., “The Omega method for discharge rate evaluation,” AIChE International Symposium on Runaway Reactions and Pressure-Relief Design, New York, 1995.
- Leung, J. C., “Easily size relief devices and piping for two-phase flow,”
*Chemical Engineering Progress,*1996. - Simpson, L. L., “Estimate two-phase flow in safety devices,”
*Chemical Engineering,*1991. - Leung, J. C., “A theory on the discharge coefficient for safety relief valve,”
*Journal of Loss Prevention in the Process Industries,*2004.

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