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# Fine-tune relief calculations for supercritical fluids

Improvement in process simulation can assist in relief load and valve sizing.

Keywords: [valve] [supercritical fluid] [pressure relief valve]

In the past 40 years, several different methods have been suggested for relief load and pressure relief valve (PRV) orifice sizing calculations for a supercritical fluid exposed to an external heat source. The following sources include some of these methods:

• API 521 suggests the use of a latent heat of 50 Btu/lb for hydrocarbons near the critical point. In the absence of a better method, this led to the use of 50 Btu/lb for even supercritical fluids.
• “A Calculation of Relieving Requirements in the Critical Region”1
• “Rigorously Size Relief Valves for Supercritical Fluids”2
• “Calculation of Relief Rate Due to Fluid Expansion and External Heat.”3

The most recent method, “Calculation of Relief Rate Due to Fluid Expansion and External Heat,” was presented at the API 2010 Summer Meeting. As the title suggests, the relief load is calculated based on the expansion of the fluid due to absorbed heat. This method can be used for any fluid, including vapor and liquid, as long as no phase change occurs. To maintain a constant pressure at a fixed volume, the relief rate at any interval must be equal to the additional volume created by the change in specific volume from heat input to the fluid. However, some assumptions must be made and some basis must be set to make this method viable:

• Other than the relieving stream, no fluid enters or leaves the vessel during the course of relief
• There is no change of phase during the course of relief.

A simple equation can be set to calculate the relief rate at each interval:

 (1)

where:
VR = Volumetric relief rate
Q = Heat input
h1 = Initial specific enthalpy
h2 = Final specific enthalpy
r1 = Initial density
r2 = Final density

The mass relief rate can be determined using the average of the initial and final densities for each interval.

 (2)

where:

MR = Mass relief rate

Both the volumetric and mass relief rates will change during the course of a relief as the specific volume and enthalpy of the fluid change. To estimate the relief rates at different intervals, one can generate a property table in a process simulator to calculate the densities and specific enthalpies of the fluid at a constant relief pressure over a given temperature range. The volumetric and mass relief rates for each interval can be calculated using Eq. 1 and Eq. 2, respectively.

In this study, a series of calculations were conducted for randomly selected n-paraffins, i-paraffins and aromatic compounds from C1 to C16, using the Peng-Robinson equation of state (EOS). The results indicate that the maximum mass relief rate occurs at lower temperature than the maximum volumetric relief. Both temperatures where the maximum relief rates occur are greater than the critical temperature. Improving the calculation precision by reducing the temperature increments does not affect the temperatures at which the mass and the volume relief rates peak. (Smaller temperature increments result in a smaller enthalpy change, ∆h, which translates to a smaller time span.) In fact, it is possible to mathematically prove that the two peaks occur at two different temperatures for real gas. This is where this article differs from the one presented at the API meeting.3

The subject was examined using two different approaches to calculate maximum relief rates (volumetric and mass) for n-hexane at 660-psia relief pressure with 5 million Btu/hr absorbed heat and a one-hour duration.

In the first approach, the relief rates were calculated by setting up property tables and using Eqs. 1 and 2 for three different temperature increments. The second approach was based on stepwise simulation models with three different time spans. The initial and final temperatures were made the same to apply the same bases for all calculations. Results are plotted in Figs. 1, 2 and 3.

The time spans in these plots are six minutes for Fig. 1, three minutes for Fig. 2, and two minutes for Fig. 3. It is clear that the impact of reducing time span on the temperatures at which the relief rates peak is insignificant. It is also obvious that the two methods yield almost the exact same results for the volumetric relief rates and very similar results for the mass relief rates. The small difference in mass relief rate is due to the fact that, in the first approach, at each interval the average of the initial and the final densities are used to convert volumetric relief rate to mass relief rate. In the second approach, only the final density is used to convert volumetric relief rate to mass relief rate.

 Fig. 1. Volumetric and mass relief rates   (10 data points).

 Fig. 2. Volumetric and mass relief rates   (20 data points).

 Fig. 3. Volumetric and mass relief rates   (30 data points).

The main objective of this exercise (and the next step in the relief valve calculation) is to size the PRV orifice area. The PRV orifice area is a function of relief valve set pressure, relief load, density and some other properties of the relieving fluid. In a scenario where a vessel or container is exposed to external heat, the fluid properties (and the relief load) vary during the course of a relief. The goal is to find the maximum required orifice area, as outlined below.

#### PRV orifice calculation

The API 520 equation for compressible gas, which is derived from an ideal gas along an isentropic path, is not a suitable method for supercritical fluids, since supercritical fluids are far from ideal gas. Instead, an isentropic mass flux expression should be used for sizing relief valves in supercritical service:

 (3)

where:
G = Mass flux
v = Fluid-specific volume
P = Fluid pressure
vt = Specific volume at throat conditions
P1 = Fluid pressure at the inlet of the nozzle

Eq. 3 is the result of a volumetric energy balance for an isentropic nozzle, and it is valid for any homogeneous fluid regardless of the non-ideality or compressibility of the fluid. Derivation details of the equation and the numerical examples for mass flux calculation are presented in Appendix B of API 520.

Eq. 3 can be solved with a numerical integration technique. With the use of a process simulator, a property table can be generated along the isentropic line to find specific volumes at various pressures, beginning at relief pressure and moving down to the relief valve back pressure. Solving Eq. 3 for each downstream pressure will result in a series of mass fluxes, which will peak when the flow is choked in the nozzle. The required orifice area for the relief valve may be simply calculated by dividing the mass flux by the mass relief rate and the discharge coefficient:

 (4)

where:
A = Required orifice area
Kd = Relief valve discharge coefficient

It is surprising that the maximum required orifice area is not in line with either the maximum mass relief rates or the maximum volumetric relief rates. Figs. 4–7 illustrate the relationship between the maximum relief rates (mass and volumetric) and the maximum required orifice area for the relief valve for some of the hydrocarbons used in this study.

 Fig. 4. Methane relief at 1,346 psia.

 Fig. 5. Iso-octane relief at 745 psia.

 Fig. 6. Hexadecane relief at 412 psia.

 Fig. 7. Benzene relief at 1,428 psia.

Fig. 8 shows the relationship between the maximum mass relief rate, the maximum volumetric relief rate, and the maximum required orifice area for n-pentane at various relief pressures. The maximum required orifice area appears at a temperature between the corresponding temperatures of the maximum volumetric and maximum mass relief rates for relief pressures from PR = 1 to PR = 7. Similar patterns were observed for other pure hydrocarbons used in the study.

 Fig. 8. N-pentane supercritical relief.

#### Numerical example

The following example illustrates relief load and orifice-sizing calculations for a vessel containing n-hexane and absorbing 5 million Btu/hr of heat with a relieving pressure of 660 psia (PR = 1.5).

Relief load calculation. A spreadsheet is used to calculate the relief rates at various stages of a relief incident. Utilizing a process simulator, a property table was created to calculate densities, along with specific enthalpies and entropies of the fluid at various temperatures.

Using Eqs. 1 and 2, the volumetric and mass relief rates are calculated at different temperatures. The relief rates will peak if the temperature range is wide enough to cover the temperatures at which the peaks occur. Table 1 is a sample calculation for n-hexane at PR = 1.5. As shown in Table 1, the maximum mass relief rate occurs when the temperature in the vessel reaches 510.9°F (TR = 1.062) and the maximum volumetric relief rate is 528.9°F (TR = 1.081).

Relief valve orifice calculation. In the process simulator, a constant entropy table has been developed for each entropy between the maximum mass and the maximum volumetric relief rates in Table 1. The property tables include the specific volume of the fluid at different pressures, from relief pressure to PRV back pressure. Using a spreadsheet, the mass flux is calculated by numerically integrating “vP” along the range of pressures, from relief pressure to the PRV back pressure. The maximum mass flux represents the choked conditions in the nozzle. Tables 2–4 show sample calculations for three different entropies.

Now the final table can be generated to calculate the maximum required orifice area throughout the relief event. Each row of the table will include throat pressure, specific entropy, mass relief rate, maximum mass flux, and the required orifice area, which is calculated from the mass relief rate and the mass flux using Eq. 4. The orifice area calculation is presented in Table 5. For a relief valve with a 0.95 discharge coefficient, the actual required orifice area would be 0.564/0.95 = 0.594 in2.

#### Takeaway

As process simulator capability increases, the ability of engineers to utilize this software allows for a significantly more precise calculation process. The possibility to generate additional data points for this calculation by decreasing the step change in enthalpy will help increase the precision of the calculation.

However, it is shown that, at extremely small step changes, the temperatures at which the maximum mass rate and maximum volume rate are generated do not approach each other. Sizing a relief device in this fashion will ensure that the orifice is adequately sized without the application of an overly conservative factor. HP

LITERATURE CITED

1 Francis, J. O. and W. E. Shackelton, “A Calculation of Relieving Requirements in the Critical Region,” API Proceedings—Refining Department, 50th Mid-Year Meeting, 1985.
2 Ouderkirk, R., “Rigorously Size Relief Valves for Supercritical Fluids,” Chemical Engineering Progress, August 2002.
3 Freeman, S., and D. Huyen, “Calculation of Relief Rate Due to Fluid Expansion and External Heat,” API Summer Meeting, 2010.

 The authors Piruz Latifi Nezami is a process engineering section manager with Jacobs Engineering in Houston, Texas. He holds a BS degree in chemical engineering from Sharif University of Technology in Tehran, Iran, and has more than 30 years of experience in the design and engineering of chemical, petrochemical and refining projects. Jerry Price is a refining and petrochemicals consultant for Jacobs Consultancy Inc. in Houston, Texas. Jacobs Consultancy provides expert consulting services to the global oil, refining and chemical industries. Mr. Price previously worked as a process engineer for Jacobs Engineering Group. He holds a BS degree in chemical engineering from Washington University in St. Louis, Missouri.

• All comments are subject to editorial review.
All fields are compulsory.

R. MacGregor
02.23.2015

Juon Wah says that Eq. 1 is wrong because is is dimensionally inconsistent. I think this is because the printers missed brackets around the (1/rho1-1/rho2) term. I've been through this article and noticed some other misprints:
1. r1 and r2 are given for rho1 and rho2 on page 1of11
2. I don't know if they didn't print or if they were missed, but the right hand side of eq. (3) should be enclosed in brackets, with the "MAX" outside of the brackets
3. The word "at" is missing in the last line above Table 1 in the text: "...volumetric relief rate is [at] 528.9F..."
4. In Table 5, the correct vol. rel. rate units are ft3/hr and the correct mass flux units are lb/s-ft2.
I really like Mr. Nezami's papers. They're well-written and interesting. But typos can create confusion, which should be cleared up. That's why I'm submitting this comment and I hope it helps other readers. If anybody sees that I've made a mistake myself here, please submit your own comment so that all readers can benefit.

Hai Nguyen Cong
09.17.2012

Hi,

"Rigorously Size Relief Valves for Supercritical Fluids" and this study are discussed within single component only. However, people prefer to designing "fully rated" for Well Head Platform in recent days. Thus, process design pressure will higher than CITHP which is normally very high. This leads to some fire case PSVs set pressure has to be equal to design pressure and higher than critical pressure of fluid. During fire, temperature increases and bring fluid to supercritical zone. In this case, we have the same challenge. Above method can be applied if we consider molecular weight of fluid is unchanged by time. Is this acceptable?

By the way, building a relationship of required relief load area to time is quite appropriate; however, evaluating required relief load area to vessel temperature has its own advantages sometimes I think.

Juon Wah
06.28.2012

Equation 1 is dimensionally inconsistent; so it can't possibly be correct.

JoeanneA
06.28.2012

This is the same calculation method we use for SuperCritical Fluids. Actually, once you know the theory, there is a way to let Hysys iterate the calculations with an embedded table function :-) But I also like to do this long method, also the longer (but not by far) method is good for those who are not yet familiar with the method as they can do it step by step : ie, flashing the streams in the simulator, then inputting the generated enthalpies and densities in a spreadsheet. BTW, I noticed that for some supercritical fluids there is a fluctuation of the mass flux (i.e. it reaches a maximum at a given temp, goes down steadily and spikes again). How do you deal with that? What I usually do is cut out the relief rate at relieving time = 1 hour as we assume that 1 hour is enough time for the fire to have been killed, and then choose the mass rate from t=0 to t=1 hour.

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