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:
V_{R} = Volumetric relief rate
Q = Heat input
h_{1} = Initial specific enthalpy
h_{2} = Final specific enthalpy
r_{1} = Initial density
r_{2} = Final density
The mass relief rate can be
determined using the average of the initial and final densities
for each interval.

(2) 
where:
M_{R} = 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 nparaffins,
iparaffins and aromatic compounds from C_{1} to
C_{16}, using the PengRobinson 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 nhexane at 660psia relief pressure
with 5 million Btu/hr absorbed heat and a onehour
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 = Fluidspecific volume
P = Fluid pressure
v_{t} = Specific volume at throat
conditions
P_{1} = 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 nonideality 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
K_{d} = 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. Isooctane 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
npentane 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 P_{R} = 1 to
P_{R} = 7. Similar patterns were observed for
other pure hydrocarbons used in the study.

Fig. 8. Npentane
supercritical relief.

Numerical example
The following example illustrates
relief load and orificesizing calculations for a vessel
containing nhexane and absorbing 5 million Btu/hr of heat with
a relieving pressure of 660 psia (P_{R} =
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 nhexane at
P_{R} = 1.5. As shown in Table 1, the maximum
mass relief rate occurs when the temperature in the vessel
reaches 510.9°F (T_{R} = 1.062) and the
maximum volumetric relief rate is 528.9°F
(T_{R} = 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 in^{2}.
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 MidYear
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.
