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Fine-tune relief calculations for supercritical fluids

06.01.2012 | Nezami, P. L., Jacobs Engineering, Houston, Texas; Price, J. , Jacobs

Consultancy, Houston, Texas

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

Keywords:

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 “v ∆P” 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.

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.