Chemical Engineering April 2013 - 57

NOMENCLATURE1
G0
Mass flux,
lb/h *
Pn
in.2
ρ Density, lb/ft3
P0
Relieving
pressure, psi
Nozzle exit
pressure, psi
A Orifice area, in.2
W Relieving mass rate,
lb/h
Kd
Discharge coefficient,
unitless
Pi Pressure at stage i, psi
ρi
Density at stage
i, lb/ft3
1. Unit conversion may be required, depending on the units selected.
(3)
The method is performed by using a
process simulator to generate data
points for the fluid density at various
pressures, utilizing an isentropic flash
routine over a pressure range from the
relieving pressure to the exit pressure.
The simulation data are used to determine
the theoretical mass flux at each
point.
Using Equation (3), the maximum
mass flux is determined by calculating
the mass flux over incrementally
larger pressure ranges, beginning at
the relieving pressure, and observing
where a maximum flux is reached. If
the maximum occurs at the relief-valve
exit pressure (built-up backpressure),
then the flow is not choked. Generally
accurate results can be obtained with
pressure increments as large as 1 psi,
but smaller step sizes can be specified
if desired [2]. Once the mass flux is
determined, the required relief valve
orifice area* can be determined from
Equation (2).
The value of the discharge coefficient,
Kd, depends on the phase of the fluid
and varies by the manufacturer of the
relief valve. The discharge coefficient
corrects for the difference between the
theoretical flow and the actual flow
through the nozzle. This value is determined
empirically for liquid and vapor
and reported by vendors for each make
and model of relief valve. If vendor
data are not available, an initial guess
of 0.975 for gases, or 0.65 for liquids
can be used [1].
For two-phase flow, the liquid-discharge
coefficient should be used if
flow in the valve is not choked and the
maximum mass flux will occur at the
relief-valve exit pressure. If the flow is
choked, then the gas-discharge coefficient
should be used and the maximum
* While relief valves are designed with a nozzle,
the area at the end of the nozzle is commonly
referred to as the " orifice area " .
mass flux will occur at some pressure
above the relief-valve exit pressure.
This is called the choked pressure [6]
Implementation
It is possible to fully automate the
Direct Integration Method using a
spreadsheet program (such as Microsoft
Excel 2010) and a process simulator
(such as AspenTech HYSYS 7.2) [7].
Users can automate the process to the
point where all they would need to do is
simply hit a button in the spreadsheet
program and the numerical integration
will be performed on an existing stream
in the simulator using a VBA (Visual
Basic for Applications) program.
First, the spreadsheet is set up to
accept the pressure and density data
for the numerical integration points.
The inlet and outlet pressure points,
pressure step size, and name of relief
stream in the simulator are placed
into specific cells in the spreadsheet,
which are referenced in the VBA code.
The VBA code instructs the simulator
to create a new, ideal expander process
block and associated streams in
the simulator. The code then iterates
across the pressure range and modifies
the pressure of the expander product
stream and automatically exports
the pressure and density data to the
Excel spreadsheet.
For each data point in the spreadsheet,
the summand, cumulative sum,
and mass flux are calculated using
Equation (3) with typical spreadsheet
formulas. When a maximum mass flux
is reached, the spreadsheet uses this
maximum flux value to calculate an
orifice size, given the relieving mass
rate and coefficients. Alternatively,
the data can be collected using the
" databook " feature in the simulator
and copied into the spreadsheet using
a simple copy-and-paste operation.
Two-phase relief scenarios
The existing single-phase vapor and
non-flashing liquid methods are relatively
easy to calculate and the resulting
predictions are fairly accurate at
conditions well away from the critical
pressure. However, two-phase models
are more difficult to implement. Existing
two-phase flow models approximate
the pressure-density relationship
of the fluid in order to calculate
the integral in Equation (3).
One of the simplest models, the
Omega Method, assumes a linear
pressure-density relationship, with
the omega parameter (ω) representing
the slope of the pressure-density
curve. An analytical solution to the
isentropic nozzle equation was developed
using the omega parameter to
solve the integral [8].
The TPHEM Method uses three
pressure-density points to define coefficients
for an empirical equation
of state " model " [9]. The empirical
equation is then used to evaluate the
integral numerically. Pressure-density
data for these models are often
provided by a process simulator. If
a simulator is available, then it
is much simpler to use the Direct
Integration Method.
The Direct Integration Method is
fundamentally different from the
other methods described here because
it does not generate an explicit equation-of-state
model to relate pressure
and density. Instead, pressure and
density data are generated using the
full thermodynamic models available
in the selected process simulator, and
these data are then used to solve the
integral numerically. Since there is
no reliance on a curve-fit pressuredensity
model, the Direct Integration
Method is more exact and reliable,
assuming the simulator's thermodynamic
model is accurate. Specifically,
there is no chance for inaccuracies
associated with the fluid equation of
state " model " propagating through the
rest of the calculations resulting in
inaccurate mass flux estimations and
ultimately an inappropriate reliefvalve
area [8, 9, 10].
Note that the Direct Integration
Method assumes that the two-phase
fluid is homogeneous, and that the
fluid is in mechanical and thermodynamic
phase equilibrium. The homogeneous
assumption is valid for most
two-phase reliefs due to high velocity
in the nozzle, which promotes mixing
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Chemical Engineering April 2013

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