Hydrocarbon Processing - August 2021 - 31

Special Focus Valves, Pumps and Turbomachinery
H. HOJJATI and B. HOJJATI, WSP Canada Inc.,
Calgary, Alberta, Canada
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. 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
G2 = −2,000 × ρt
2 × ∫
dP
P ρr
equation (Eq. 1), which relates mass flux (G) at the nozzle throat
to the fluid density (ρ) and the pressure (P) variations:
Pt
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
is then corrected using a dimensionless constant known as discharge
coefficient (Kd
), 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
277.8 × m !
A =
G × ∏Ki
where ∏ Ki
including discharge coefficient, backpressure (Kb
and combination (Kc
varies between relief pressure (Pr
Pr < Pt
≤ Pb
(2)
is the product of all applicable correction coefficients,
), viscosity (Kv
)
) correction factors. The throat pressure
) and backpressure (Pb
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 inHydrocarbon
Processing | AUGUST 2021 31
) in Eq 3:
(3)
The theoretical mass flux
(1)
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