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Published in Chemical Enginering, 112, no. 9, pp 42-50, Sept, (2005)
Page 1 15 June 2005
PROPERLY SIZE SAFETY RELIEF SYSTEMS FOR ANY CONDITIONS
Ron Darby
Professor Emeritus
Department of Chemical Engineering
Texas A&M University
College Station, TX 77843-3122
(979) 846-4175 Fax: (979) 846-4175
r-darby@tamu.edu
Overview
Safety relief valves and rupture disks are typically used to protect equipment from excessive
overpressure. Typical scenarios that can result in such overpressure in excess of the vessel
MAWP (maximum allowable working pressure) include external fire, blocked outlet line, power
failure, loss of cooling water or steam, thermal expansion, excess inlet flow, accumulation of
non-condensables, failure of check or control valve, exchanger tube rupture, runaway reaction,
human error (e.g. open or close the wrong valve), etc. These and other scenarios are discussed in
more detail by Wong [1].
Reliefs should be installed on all vessels other than steam generators, including reactors,
storage tanks, towers, drums, etc. Other locations where reliefs are required are blocked in
sections of liquid filled lines exposed to external heating, the discharge from positive
displacement pumps, compressors and turbines, and vessel steam jackets. Storage vessels
containing volatile liquids and a vapor space should be protected from both excessive pressures
from external heat or flow input but also from the possibility of a vacuum from condensation of
the vapor.
Relief valves are designed to open at a preset pressure and are sized to allow mass flow
out of the vessel at a rate sufficient to remove excess energy from the vessel at least as fast as it
is input to the vessel contents (from either external or internal sources, e.g. external heating,
Published in Chemical Enginering, 112, no. 9, pp 42-50, Sept, (2005)
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runaway reaction, etc.) to prevent further pressure build up. The valve will close when the
pressure drops to a safe level, thus containing and protecting the bulk of the vessel contents.
Since the capacity of a valve is limited, it cannot accommodate the extreme flow rate that might
be required to protect against an extremely high energy input rate such as might result from a
very energetic runaway reaction, a deflagration or explosion. Rupture disks are a less expensive
alternative, especially for very large capacity requirements, but of course do not reclose to
contain the vessel contents.
Proper design of a relief system requires not only determining the correct size for the
valve or rupture disk, but also the proper size and selection of upstream and downstream piping
and effluent handling systems. The procedure for the design for a safety relief system can vary
from a relative simple, fairly routine, process for single phase (gas or liquid) flow to a complex
procedure for two-phase flow requiring considerable expertise and procedures that depend on
conditions and the nature and characteristics of the fluid being discharged. The details of this
total process are beyond the scope of this article, and authoritative references should be
consulted for further details and procedures (e.g. API [2, 3], CCPS [4], DIERS [5]). This article
will summarize the overall considerations important in the design process, and will concentrate
on the basic procedure for properly sizing the relief (either a SRV or rupture disk) under a
variety of conditions for single and two-phase flows.
Required Relief Rate
The first step in the design process for valve sizing is to postulate one or more credible scenarios
that could result in unacceptable overpressure, and determine the corresponding required
discharge mass flow rate
()
m
that would be sufficient to prevent overpressure in excess of the
vessel MAWP. The required relief mass flow rate ( m
) is determined by an energy and mass
Published in Chemical Enginering, 112, no. 9, pp 42-50, Sept, (2005)
Page 3 15 June 2005
balance on the vessel under the conditions of the specific postulated relief scenario, e.g. a
runaway reaction, external fire, loss of cooling, blocked line, etc. The value of m
is determined
by the requirement that the rate of energy discharge from the vessel be equal to or greater than
the maximum rate at which excess energy is input into the vessel under the assumed scenario. It
is normal to postulate a “worst case” scenario and a “most credible” scenario, and base the
design on the worst of the likely cases. This is, of course, a judgment call coupled with the
probability of the scenario occurring.
For a runaway reaction involving volatile or gaseous components, data from an adiabatic
calorimeter or detailed kinetic information are required to predict the required relief rate.
Specialized techniques and/or equipment are needed for this, and the process should be left to the
experts (see e.g. DIERS [5], CCPS [4], Darby [6]).
For storage vessels containing a volatile liquid, a common scenario is an external fire
which heats the vessel and contents, resulting in superheating the liquid. If the vapor pressure
builds up to a point which exceeds the vessel MAWP, the vessel could rupture resulting in a
BLEVE (Boiling Liquid Expanding Vapor Explosion). The relief mass flow rate must be high
enough that the rate of discharge of the total sensible and latent heat through the vent must equal
or exceed the rate of heat energy transferred to the fluid through the vessel wall from the fire
exposure. Since the liquid will typically be superheated, flashing will occur as the pressure
drops through the vent; resulting in two-phase flow in the relief which must be accounted for in
sizing the relief, as described below (the relief area required for two-phase flow is normally
significantly larger than that which would be required for single phase flow). Methods for
estimating the heat transfer rate from a fire to storage vessels are presented by NFPA [7] and API
Published in Chemical Enginering, 112, no. 9, pp 42-50, Sept, (2005)
Page 4 15 June 2005
[8]. For conditions not adequately covered by these documents, Das [9] has presented
fundamental relations for determining the heat transfer rate.
Valve Sizing
The required orifice area1 for a relief valve or rupture disk is determined from the formula
do
m
AKG
= (1)
Here m
is the required relief mass flow rate (mass/time) and o
G is the theoretical mass flux
(mass/time area), calculated for flow through an ideal (isentropic) nozzle. The expression for
o
G follows directly from application of the general steady state energy balance (Bernoulli)
equation to the fluid (gas, liquid or two-phase) in the nozzle (see e.g. Darby [10]):
n
1
1/ 2
P
on
P
dP
G2
⎛⎞
=ρ −
⎜⎟
⎜⎟
ρ
⎝⎠
∫ (2)
where 1
P is the pressure at the entrance to the valve, n
P is the pressure at the nozzle exit,
ρ
is the
fluid (or mixture) density at pressure P, and n
ρ
is the fluid density at pressure n
P, the nozzle exit
or throat.
d
K in Eqn (1) is the (dimensionless) discharge coefficient that accounts for the difference
between the predicted ideal nozzle mass flux and the actual mass flux in the valve. This is
determined by the valve manufacturer from measurements using (typically) single-phase air or
water flows. Further assumptions must be made to determine the appropriate value of d
K to use
for two-phase flow (this is discussed later).
1 Although the term “orifice” is commonly used to describe the minimum flow area constriction in the valve, the
geometry more commonly resembles a nozzle and the area is determined by applying the equation for flow in an
isentropic nozzle, as described in this article.
Published in Chemical Enginering, 112, no. 9, pp 42-50, Sept, (2005)
Page 5 15 June 2005
There are a finite number of standard valve nozzle (orifice) sizes to choose from and the
calculated area A would not be expected to correspond exactly to one of these sizes. In practice,
a 10% “safety factor” is automatically applied to the calculated area (per the ASME code), and
then the standard size nozzle orifice area which is the closest to the resulting value on the high
side is then selected.
It is important that the relief area be neither too large nor too small. An undersized vent
would obviously not provide the required overpressure protection, whereas an oversized vent
will result in excessive flow which can adversely affect the opening and closing characteristics of
the relief valve resulting in impaired performance (e.g. unstable operation, or chatter) with
possible severe damage to the valve. If the valve is oversized, the actual flow rate will be
significantly greater than the required design rate
(
)
m
so that if the associated piping is sized for
the design rate it will be undersized for the actual rate. This means the pressure drops through
the entrance and exit piping will be greater than expected, and these pressure drops can have
serious adverse effects on the stability of the valve (see the section on Inlet and Discharge
Piping).
Although the flow through a relief valve is an unsteady (time-dependent) process, it is
customary to base the calculations on assumed steady state conditions corresponding to the
expected flow rate at a pressure which is 110% of the relief set pressure (i.e. 10% overpressure).
The relief set pressure is normally the vessel MAWP, although other relief pressures are allowed
by the ASME code for various special cases (e.g. API [2]).
Nozzle models
The term “model” as applied to valve sizing is frequently misunderstood. For example, the
commonly referenced “Homogeneous Equilibrium Model (HEM)” is not a “complete model” for
Published in Chemical Enginering, 112, no. 9, pp 42-50, Sept, (2005)
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calculating the nozzle mass flux, but simply a set of conditions and assumptions which constrain
the calculations. The HEM implies that if the fluid through the valve is a two-phase gas-liquid
mixture, it will be sufficiently well mixed that it can be described as a single-phase fluid with
properties that are a suitable combination of those each fluid, and that the two phases are in both
mechanical and thermodynamic equilibrium. These assumptions are necessary, but not
sufficient, for calculating the nozzle mass flux because additional assumptions or conditions
must be specified with regard to the properties of the fluid which are necessary to determine the
mixture density as a function of pressure. It is evident from Eqn (2) that the calculated nozzle
mass flux is determined specifically by the manner in which the fluid density depends on
pressure over the range of pressures in the nozzle. The “homogeneous equilibrium assumption”
is inherent in the derivation of Eqn (2), but the specific relation to be used for the
()
Pρ function,
and the manner in which the integral is evaluated using this function, must also be specified for
the “model” to be complete.
Single-Phase Liquid Flow - For single-phase liquid flow, the nozzle mass flux integral (Eqn 2)
is simple to evaluate since the fluid density is assumed independent of pressure. Thus, for
liquids with a constant density, Eqn (2) reduces to
()
oon
G2PP=ρ − (3)
This equation is valid for fully turbulent flow (e.g Reynolds numbers above about 100,000), for
which the flow rate is independent of the fluid viscosity. For low Reynolds number (e.g. high
viscosity) flows, the value given by Eqn (3) can be multiplied by a correction factor
(
)
v
K that
reflects the dependence of o
G on Reynolds number as well as on the d/D = β ratio of the nozzle
(Darby and Molavi [11]):
Published in Chemical Enginering, 112, no. 9, pp 42-50, Sept, (2005)
Page 7 15 June 2005
()
0.1
v1.4
Re
K 0.975
950 1 / N 0.9
β
=⎡⎤
−β +
⎣⎦
(4)
where Re
N is the Reynolds number through the nozzle. For a two-phase mixture, a volumetric
average values of density and viscosity is used. Here, d/D
β
=, the ratio of the nozzle diameter
to valve inlet diameter. These equations also assume that the liquid is Newtonian. There are no
known data for non-Newtonian flow in relief valves and there are no current models that account
for such properties. However, in the absence of more specific information, it may be assumed
that Eqn (4) can be applied to non-Newtonian viscous fluids if the Reynolds number is modified
accordingly for the specific non-Newtonian rheological model (see e.g. Darby [10], Ch 7).
Single-phase gas flow – In the case of an ideal gas, the integral can be readily evaluated
assuming isentropic flow for which k
P/
ρ
= constant, where k is the isentropic exponent (which
for an ideal gas is the ratio of the specific heat at constant pressure to that at constant volume).
However, the result depends upon whether or not the nozzle exit pressure
()
n
P is at or below the
value at which the speed of sound is reached in the nozzle (i.e. choked flow). The criterion for
choked flow is nc
PP≤, where
()
(
)
k/ k 1
co
PP2/k1
−
=+⎡⎤
⎣⎦
. If the flow is choked the mass flux is
given by
(
)
(
)
k1/2k1
ooo
2
GkP
k1
+
−
⎛⎞
=ρ
⎜⎟
+
⎝⎠ (5)
which is independent of the downstream pressure. If the flow is not choked (i.e. sub-critical),
nc
PP>and the mass flux depends on both the upstream and downstream pressures as follows:
()
1/2
2/k k 1 /k
oo nn
o
oo
2P k PP
Gk1 P P
+
⎡
⎤
⎛⎞ ⎛⎞
ρ
⎢
⎥
=−
⎜⎟ ⎜⎟
−
⎢
⎥
⎝⎠ ⎝⎠
⎣
⎦
(6)
Published in Chemical Enginering, 112, no. 9, pp 42-50, Sept, (2005)
Page 8 15 June 2005
If on
(P / P ) 2≥ (approximately), the flow will probably be choked and Eqn (5) applies. Non-ideal
gases can be treated using Eqn (2) along with actual property data or an appropriate equation of
state to evaluate the gas density. Alternately, the above equations can be used if a “non-ideal k
value” is used, and the density is divided by an appropriate value of the compressibility factor (z)
evaluated at the choke conditions (see Shakleford [12] for a discussion of the suitability of using
ideal vs non-ideal gas k values).
Two-phase flow
Thousands of relief valves in process plants are installed in vessels that operate under conditions
that can result in two-phase flow through the relief valve, and the valve must be properly sized to
accommodate such flows. Various conditions could result in flashing, condensing, or “frozen”
(non-flashing) flow. Flashing flow occurs in nozzles/valves whenever the entering fluid is a
saturated liquid, a sub-cooled liquid that reaches the saturation pressure within the nozzle, or a
two-phase vapor-liquid mixture. Frozen two-phase flow may occur if the vessel initially
contains both gas and a non-volatile liquid (e.g. a vessel with inert gas padding). Either frozen or
flashing flow could result from a runaway reaction, for example. Retrograde condensation may
also occur when the fluid in the vessel is a dense gas that condenses when the pressure drops.
Two-phase flow is considerably more complex than single-phase flow, and there are a
number of additional factors that must be considered such as the flow regime, the nature of the
interaction between the phases, the method of determining the properties of the two-phase
mixture, and the method of incorporating these properties into evaluation of the mass flux.
If a vessel initially contains both liquid and gas or vapor, or a superheated liquid, the
mass fraction of gas (i.e. the quality) in the two-phase mixture entering the relief device will
depend upon the amount of gas/vapor generated within the liquid phase, the degree of mixing in
Published in Chemical Enginering, 112, no. 9, pp 42-50, Sept, (2005)
Page 9 15 June 2005
this phase, the bubble rise velocity, the physical properties of the liquid, and the initial void
fraction ( i.e. the vapor space) in the vessel. The prediction of this initial quality can be a
complex procedure, and the pertinent references should be consulted (e.g. CCPS [4].
Flow Regime: This refers to the distribution of the two phases in the flow field, which can be
classified as distributed (such as stratified, wavy, slug, or bubbly) or homogeneous (i.e. well
mixed). Because of the high velocities and high degree of turbulence in typical relief flows, the
usual assumption is that the flow is well mixed and hence homogeneous within the relief device.
This means that the two-phase mixture can be represented as a “pseudo single-phase” fluid, with
properties that are a suitable average of the individual fluid properties. There are many ways that
this average can be defined, but the most widely accepted is a volume-weighted average. On this
basis, the density of the two-phase mixture is given by
(
)
GL
1
ρ
=αρ + −α ρ (7)
where α is the volume fraction of the gas phase, given by
()
GL
x
xS1x /
α=
+
−ρρ
(8)
Here x is the quality (i.e. mass fraction of the gas phase) and S is the slip ratio, i.e. the ratio of
the gas velocity to the liquid velocity in the mixture (see below).
Mechanical Equilibrium: This implies that the two phases are flowing at the same velocity, i.e.
no slip (S = 1). Slip occurs because the gas phase expands as the pressure drops and hence must
speed up relative to the liquid phase. Slip becomes more important as the pressure gradient
increases, and is most pronounced as the velocity approaches the speed of sound (choking).
Although there are a variety of “models” in the literature for estimating slip as a function of fluid
properties and flow conditions, it is often neglected under pressure relief conditions because of
the high degree of turbulence and mixing. For flashing flows, slip effects are normally
Published in Chemical Enginering, 112, no. 9, pp 42-50, Sept, (2005)
Page 10 15 June 2005
negligible, since the volumetric expansion due to flashing will overwhelm the expansion of the
gas phase due to pressure drop alone. However, slip can be significant for frozen flows (e.g. air
and cold water). For example, Jamerson and Fisher [13] and Darby et al. [14] found that a slip
ratio (S) of 1.1 to 1.5 is consistent with various frozen flow data in nozzles. Most frozen flow
data in the literature are for air-cold water mixtures, and there are little or no data for industrial
fluids. Note that Eqn (8) shows that an increase in S results in a larger two-phase density and
corresponding higher mass flux than would be predicted with no slip. Some models for the
nozzle mass flux include provision for slip and some do not, as described later.
Thermodynamic Phase Equilibrium: It is commonly assumed that the gas or vapor phase is in
local thermodynamic equilibrium with the liquid phase, which means that the properties of the
mixture are a function only of the local temperature, pressure and composition. In other words,
when the pressure in a liquid drops to the saturation (vapor) pressure, vaporization (flashing) will
occur instantly if the system is in equilibrium. However, flashing is actually a rate process that
takes a finite time (e.g. a few milliseconds) to develop fully. During this “relaxation time” a
liquid can travel several inches (i.e. a corresponding “relaxation distance”) in the nozzle of a
valve under typical relief conditions. Under these conditions the amount of vapor generated (e.g.
the quality) is much smaller than would occur under equilibrium conditions, and the mixture
density and mass flux are correspondingly larger. Experimental data on a number of single
component systems (e.g. Henry and Fauske [15]) have indicated that this “relaxation length” is
of the order of 10 cm for typical relieving conditions, which means that flashing flow in nozzles
shorter than 10 cm should be in non-equilibrium. Some nozzle flow models have provision for
non-equilibrium effects and some do not, as discussed later.
Published in Chemical Enginering, 112, no. 9, pp 42-50, Sept, (2005)
Page 11 15 June 2005
Thermodynamic Path: As the fluid flows through the nozzle, the pressure and temperature drop
and the volume fraction of gas/vapor increases. For frozen flows, the mass flow rate of each
phase remains constant throughout the flow path, although the phase volume fractions change
because the gas expands. For a volatile liquid, the quality (i.e. the mass fraction of gas) will also
change from point to point because of increasing evaporation as the pressure drops, and it is
necessary to determine the local quality as a function of pressure in order to calculate the two-
phase mixture density from Eqn (7). This is done by assuming the fluid follows a specific
thermodynamic path as it traverses the nozzle, which may be isothermal, isentropic, or
isenthalpic, and then determining the gas and liquid densities and the quality (or phase ratio)
along this path. The usual assumption is that this path is isentropic, since the “isentropic nozzle
equation” is used as the basis for the mass flux. On the other hand, a case can be made for
assuming that the flow in the nozzle is isenthalpic and using an “enthalpy balance” to determine
the local properties. In some cases (e.g. liquid flow), the isentropic, isenthalpic and isothermal
paths are virtually identical. For example if the inlet conditions are subcooled or saturated, and
are sufficiently far from the critical point, there is usually a negligible difference between the
isentropic and isenthalpic paths. However, as the critical point is approached, or for low
vapor/liquid ratios (low quality), the difference is more pronounced. Particularly in the vicinity
of the thermodynamic critical point the differences may be quite significant. There are no
definitive studies to show which assumption is the most appropriate, but the general consensus
favors the isentropic path (which is inherent in the isentropic nozzle equation).
Physical Property Data: In order to calculate the two-phase density (and other properties) along
the chosen path (i.e. isentropic), a database of thermo-physical properties of the fluids is
required. The specific properties and the amount of data required depend on the particular model
Published in Chemical Enginering, 112, no. 9, pp 42-50, Sept, (2005)
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used, but at a minimum the mass fraction of the gas phase (quality) and the densities of each
fluid phase are required as a function of pressure along the path (e.g. for the HDI model). For
frozen flows, the liquid density is constant so the only property data required is a suitable
equation of state for the gas (e.g. the ideal gas law), or appropriate data for the gas. Some
models require enthalpies, entropies, densities, heats of vaporization and specific heats at one or
more conditions. For flashing pure components, the required data are usually available in a
thermo-physical property database or simulator. The Omega and HNE models require thermo-
physical properties at only one state (e.g. the stagnation state), and employ an entropy or
enthalpy balance to determine the vapor fraction (quality) of the two-phase flashing mixture.
The API version of the Omega method for mixtures requires thermo-physical properties at two-
states instead of evaluation of the Omega parameter at one state. For multi-component mixtures,
additional property data or mixture models must be available and can be used with a flash routine
to determine the vapor-liquid equilibrium properties (e.g. density and quality) of the two-phase
multi-component mixture as a function of pressure.
Model assumptions – The assumptions made with regard to the above considerations constitute
the “model” for the nozzle mass flux. The most common assumption is the HEM (Homogeneous
Equilibrium Model), which implies that the two-phase mixture is homogeneous and the phases
are in equilibrium (both mechanical and thermodynamic). Several versions of the HEM are in
use, which differ in the specific assumptions and methods used to evaluate the two-phase density
and the mass flux integral (Eqn 2). Some of these variations are described below.
The Omega method - This method (Leung [16, 17]) was derived for a single component fluid,
and assumes that the density of the two-phase mixture can be represented by a linearized
equation of state. It requires fluid properties at only one state (the saturation or stagnation state).
Published in Chemical Enginering, 112, no. 9, pp 42-50, Sept, (2005)
Page 13 15 June 2005
Factors that should be considered when using the Omega method are:
• The equations are based on an analytical evaluation of the mass flux integral,
using an approximate linearized two-phase equation of state for the fluid
density. The equations are fairly complex, so care is required to insure the
calculations are correct.
• Fluid property data are required at only one state, simplifying the required
amount of input property data. However, these thermodynamic and physical
property data must be accurate, since small variations or errors in the
thermodynamic properties can have a large effect on the resulting density
values.
• The linearized equation of state may not give accurate two-phase density
values vs pressure for some conditions since it extrapolates the two-phase
density from the relief (stagnation pressure) state. The accuracy depends not
only on the nozzle conditions and the nature of the fluid but also the range of
pressures in the nozzle.
• The method tends to be unreliable in the vicinity of the critical point or for
dense gases that condense when the pressure is reduced (retrograde
condensation).
• It was derived for single component fluids and is not easily adapted to multi-
component mixtures unless modified (see the API Method below) or unless
the boiling range of the mixture is small. Consequently, it is inappropriate for
mixtures with light gas components (e.g. hydrogen).
• Neither slip nor non-equilibrium effects are accounted for in the model.
• A special version of the basic model is required for slightly subcooled liquids.
API method: The method presently recommended by API 520 [2] is the Omega method for
single component fluids and multi-component mixtures with a normal boiling range less than
150oF. The heat of vaporization is calculated as the difference between the vapor and liquid
specific enthalpies of the mixture. For flashing mixtures with a normal boiling range greater than
150oF, the ω parameter is determined from the calculated two-phase density of the mixture at
two pressures ( o
P and 9o
P0.9P=) and constant entropy. Factors to be considered when applying
this method include:
• It is basically a two-point linear fit of the two-phase density at pressures o
P
and 9
P . This is better than the one-point Omega extrapolation but still may
not give accurate results depending on the fluid, the conditions, and the
pressure range involved (particularly near the critical point).
Published in Chemical Enginering, 112, no. 9, pp 42-50, Sept, (2005)
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• The choke pressure is estimated using the “single point ω method”, which
could introduce some error or uncertainty.
• Since the two-phase density is calculated from a fluid property database at
two points using the single component thermodynamic properties, it can be
used for multi-component mixtures if an appropriate property database is
available.
• A reliable property database must be used to determine the two-phase density
and quality (x) at two separate pressures at constant entropy. For multi-
component systems, this can be done using a flash routine coupled with an
appropriate fluid database in a simulator. Accurate thermo-physical property
(density) data are required since small variations or errors in the
thermodynamic properties can have a large effect on the resulting density
values
• Non-equilibrium effects (either thermodynamic or mechanical) are not
included.
TPHEM – This model (the Two-Phase Homogeneous Model) is implemented using a computer
routine that is available on a CD that accompanies the CCPS Guidelines book “Pressure Relief
and Effluent Handling Systems” [4]. The mass flux integral (Eqn 2) is evaluated numerically by
the program using input data for the densities of the liquid and gas/vapor and the mixture quality
at two or three states at constant entropy from the stagnation pressure to the discharge pressure.
The density data are fitted in the program by an empirical equation, which is used to interpolate
the densities at intermediate pressures for evaluation of the integral. The user can choose from a
variety of empirical equations for fitting the two-phase P,
ρ
data, with one, two or three
parameters (Simpson [18, 19]).
The densities of the gas and the liquid and the quality (x) of the mixture at each of the
two or three pressures along an isentropic path are input into the program. The single parameter
density model is equivalent to the Omega method. The 2-parameter model is equivalent to the
API method, with 2o
P0.9P=. For flashing of an initially sub-cooled liquid, the three pressures
are the saturation pressure, the nozzle exit pressure, and one intermediate pressure. It is
necessary to have an accurate property database for the fluids in order to determine the required
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input density data. The program output is the mass flux at the specified exit pressure (or vice
versa). A variety of other output options are also available including viscous or non-viscous
flow, pressure drop in straight pipe with or without fittings, etc. The choke pressure and
corresponding mass flux are determined by initially specifying the stagnation pressure as the
backpressure and then decreasing this pressure in increments until the mass flux reaches a
maximum.
Some key characteristics of this method are:
• It is applicable to frozen or flashing flows, as well as subcooled or saturated
liquids.
• The program does all of the calculations automatically, so it is quick and easy
to implement.
• Two or three
()
P, , xρ data points are required along an isentropic path. Using
more than one data point can improve the property estimates considerably
over that of the Omega method in many cases. Accurate thermo-physical
property (density) data are required since small variations or errors in the
thermodynamic properties can have a large effect on the resulting density
values.
• A wide variety of conditions, including pipe flow or nozzle flow for inviscid
or viscous fluids can be run using various combinations of “switches” in the
program, for calculating either the mass flux or the exit pressure.
• It has the capability of including a slip parameter or a non-equilibrium
parameter, but there are no guidelines for selecting the values of these
parameters.
• Multi-component systems can be handled using a flash routine to generate the
required
()
P, , xρ data points if a suitable property database is available.
• Multiple runs are required in order to determine the choke pressure and
maximum (choked) mass flux.
• The multiple combinations of program “switches” and options required to
run the various cases can sometimes be confusing and requires care to ensure
proper implementation.
• The results can be sensitive to the choice of conditions for the input data and
the range of pressures required, especially near the critical point.
• Because of the density-pressure fitting equation, the choke point may not be
accurately predicted.
HNE mode: This model (the Homogeneous Non-Equilibrium model) is based on an energy
balance on a flashing liquid. It is an extension of the Equilibrium Rate Model (ERM), which
Published in Chemical Enginering, 112, no. 9, pp 42-50, Sept, (2005)
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employs an isenthalpic energy balance on a saturated liquid to determine the fraction that is
flashed. The rapid generation of vapor from the flash is assumed to result in choked flow and the
mass flux is evaluated from the definition of the speed of sound using the Clausius-Clapeyron
equation to relate the vapor density to the vapor pressure of the flashing fluid and the
thermodynamic properties (e.g. heat of vaporization, etc). The mass flux predicted by this model
for a saturated flashing liquid is typically about 10% higher than corresponding values predicted
by the HEM. However, for slightly sub-cooled liquids it has been observed that the actual mass
flux may be as much as 300% greater than predicted by either model.
This model was extended by Henry and Fauske [15], and later by Fauske [20], to account
for non-equilibrium effects resulting from delayed flashing by the rate processes involved. The
model determines the gas mass flux and liquid mass flux separately using the respective single
phase discharge coefficients dG
K anddL
K , and combines these in proportion to the respective
phase mass fractions. Non-equilibrium is characterized by a delayed flashing parameter which is
a function of the “relaxation length”, e
L10cm
=
. Non-equilibrium conditions were found to
occur when e
LL<, and equilibrium occurs if e
LL>. Factors which should be considered when
using the HNE model include:
• It is applicable to single-phase (liquid or gas), subcooled or saturated liquid, or two-phase
mixtures. It is applicable to flashing but not condensing flow conditions.
• It predicts effects of non-equilibrium conditions (e.g. in short nozzles).
• It requires property data only at the saturation state. This minimizes the amount of input
data required, but may result in lower accuracy relative to those methods that utilize data
at more conditions. Accurate thermo-physical property data are required since small
variations or errors in the thermodynamic properties can have a large effect on the
resulting density values
• The calculations are simple and easy to perform
• The choke pressure is assumed to be the saturation pressure, but this is not always
appropriate especially for low relief pressures and low subcooling. Better results may
sometimes be obtained if the actual choke pressure is used instead of
b
P in the model
Published in Chemical Enginering, 112, no. 9, pp 42-50, Sept, (2005)
Page 17 15 June 2005
equations, but this pressure has to be determined using another method (such as TPHEM
or HDI).
• The assumption of an ideal gas phase is made for the gas phase, which can introduce
errors particularly in the vicinity of the critical point.
• The model does not include any provision for slip.
• The relaxation flow length (10 cm) is based on a relatively small number of observations.
The HDI method: This method (the Homogeneous Direct Integration method, Darby et.al [21],
Darby, [22]) involves generating multiple
(
)
P, , x
ρ
data points over an isentropic range of
pressures from o
P to n
P using a thermodynamic property database for a pure fluid, and a flash
routine for a multi-component mixture. These data are used to evaluate the mass flux integral,
Eqn (2), by direct numerical integration. This can be done easily on a spreadsheet by a simple
trapezoidal rule, or a simply quadrature formula, as follows:
(9)
Pressure increments of 1 psi are usually quite adequate to provide sufficiently accurate results.
The choke point is determined by repeating the calculations at successively lower values of n
P,
starting at o
P , until the mass flux reaches a maximum. If no maximum is reached before nb
PP
=
the flow is not choked. The method is perfectly general, and applies to any fluid, under any
conditions (single-phase gas or liquid, or two-phase) for which property data are available.
This method can be extended to account for non-equilibrium effects for flashing flow in
short nozzles (L < 10 cm) (i.e the HNDI or Homogeneous Non-Equilibrium Direct Integration
model), as follows. The effect of non-equilibrium is to delay the development of flashing to a
pressure below the normal equilibrium saturation pressure. That is, when the pressure reaches
the saturation pressure the flashing process is not completely developed so that the quality (x) is
nn
o
o
1/2 1/2
PP
i+1 i
on n
Pi
P
P-P
dP
G=ρ-2 ρ-2
ρρ
⎛⎞
⎡
⎤
⎛⎞
≅
⎜⎟
⎢
⎥
⎜⎟
⎜⎟ ⎝⎠
⎣
⎦
⎝⎠ ∑
∫
Published in Chemical Enginering, 112, no. 9, pp 42-50, Sept, (2005)
Page 18 15 June 2005
actually lower than it would be under equilibrium flashing conditions
()
e
i.e. x . Since the
equilibrium two-phase density is related to the quality by
(
)
GL
1x
1x
ρρ ρ
−
=+ (26)
the density (and hence the mass flux) would be higher under non-equilibrium conditions than at
equilibrium. Thus the effect of non-equilibrium can be accounted for by appropriately
modifying the value of the quality, x. As indicated from the HNE model, observations have
shown that for typical flashing flows in nozzles equilibrium is reached at a distance of about 10
cm along the nozzle, with non-equilibrium conditions prevailing for L < 10 cm. Thus, if we
assume that x approaches the equilibrium quality e
x as L approaches 10 cm. Thus for
L10cm≤the effective quality at the nozzle throat can be estimated as
()
oeo
L
xx x x
10
=+ − (27)
where L is the nozzle length in cm, o
x is the initial quality of the fluid entering the relief device.
For L > 10 cm, e
xx=. Considerations appropriate to this method include:
• The method is rigorous within the assumptions inherent in the ideal nozzle
equation and the HEM assumptions, and the precision of the property data.
• It is universally applicable for all fluids under any/all conditions for which the
property data are available.
• The procedure does not depend on whether the entering fluid is cold liquid, sub-
cooled flashing liquid, a condensing vapor, or a two-phase mixture, or on whether
or not the flow is choked.
• It is simple to understand and apply.
• It is easily applicable to multi-component systems, provided the mixture property
data are available for performing the required flash calculations over the pressure
range of interest. A process simulator using the property database can usually
generate the required data.
• The calculation method is simple and direct, and is ideally suited to a spreadsheet
solution.
• The method is more accurate than those above because no “model approximation”
for the fluid properties is involved.
Published in Chemical Enginering, 112, no. 9, pp 42-50, Sept, (2005)
Page 19 15 June 2005
• The method can easily be applied to short (non-equilibrium) as well as long
(equilibrium) nozzles
• Accurate thermodynamic and physical property data,
(
)
Pρ, are required to give
good results.
• A flash routine must be used for multicomponent mixtures to generate the
()
P, , xρ data required for the integration, and more data points must be
computed.
• Slip effects can be readily incorporated into the method via Eqn (8) provided an
appropriate value for the slip ratio (S) is known or can be predicted.
The discharge coefficient
The discharge coefficient ( d
K ) in Eqn (1) corrects for the difference between the flow predicted
by the ideal isentropic nozzle model and that in an actual valve. Thus the values of d
K depend
upon how accurately the theoretical isentropic nozzle “model” represents the real valve flow rate.
Thus, the value of d
K depends upon both the nature (geometry) of the valve as well as the
accuracy of the fluid property “model”. Values of the gas phase coefficient dG
K are always closer
to unity (i.e. a perfect model) than the liquid phase coefficient dL
K values. This is because the gas
flow coefficients are measured under choked flow conditions, for which the isentropic ideal gas
model is a much better representation of the actual flow. Conditions for liquid flow coefficients
are obviously determined under non-choked flow conditions, for which the entire valve (not just
the nozzle) influences the flow rate and therefore the “isentropic nozzle model” is much less
adequate. Values of d
K for valves and rupture disks are determined by the manufacturer in a
certified calibrated test facility using water or air (sometimes steam), and are updated annually in
the “Red Book”2. The “Red Book”2 value or “ASME d
K ” is based on the actual area and should
be used if the ASME relief valve orifice size (actual area) is used. Values of the single-phase
2 Pressure Relief Device Certifications, National Board of Boiler Inspectors,
(http://www.nationalboard.org/Redbook/redbook.html)
Published in Chemical Enginering, 112, no. 9, pp 42-50, Sept, (2005)
Page 20 15 June 2005
d
K’s are also given in API Standard 526 “Flanged Steel Pressure Relief Valves” [23] which are
based on “standardized” nozzle (orifice) areas, as opposed to the actual area. Specifying the API
“standardized” nozzle sizes (with the corresponding values of d
K ) provides a uniform method for
sizing valves independent of the specific vendor or valve dimensions. The d
K values published
by vendors for use with the API standard orifice sizes should only be used with these sizes (API,
[2]). In general, the API Kd‘s are about 10% higher than the ASME (Red Book) d
K values, and
the API standard areas correspondingly smaller. The API values for spring-loaded relief valves
are approximately 2% higher than the ASME valves. (the product d
K A is approximately the
same for either the ASME or the API values). Use the API d
K when API standard size relief
orifice sizes are specified and the ASME d
K when the actual nozzle sizes are used.
For two-phase flow there are no validated databases or certified test facilities, so
experimental values of d
Kare not available. The few suggestions available in the literature are
based on a limited number of experimental observations. Some investigators suggest various
averaging methods for the two-phase d
K , such as a volume-weighted average of the liquid and
gas phase coefficients based on the relative volumes of liquid and gas. However, data on frozen
air/water flows in various relief valves (Darby, [22]) indicate that when a rigorous mehtod like
the HDI or HNDI is used, a value of ddG
KK
=
is appropriate when the flow is choked, and
ddL
KK= if the flow is not choked. This is quite logical, because measured dG
K values are
representative of choked flow conditions (for which the mass flux is independent of conditions
downstream of the nozzle) and measured dL
K values are representative of non-choked conditions
(where the mass flux is affected by the flow resistance in the body of the valve as well). At the
point where the transition from choked to non-coked flow occurs the pressure is discontinuous
Published in Chemical Enginering, 112, no. 9, pp 42-50, Sept, (2005)
Page 21 15 June 2005
and the flow resistance shifts from the nozzle only to the entire valve including the body
resistance. This increased flow resistance causes a corresponding reduction in the mass flux,
which is therefore also discontinuous at this point. This is also the reason that values
of dG dL
KK>, i.e. the choked flow condition under which dG
K is determined is more accurately
represented by the isentropic nozzle model, which does not include the valve body effects that
influence the value of dL
K . Since two-phase flashing flows choke much more readily than
single-phase gas flows (e.g. choking can occur at pressures as high as 90% of the upstream
pressure), it is very unusual to encounter subcritical (non-choked) conditions with two-phase
flows. Thus the use of dG
K is generally appropriate for two-phase flows.
Balanced bellows relief valves utilize a backpressure correction to account for the action
of the bellows in compensating for the backpressure and enhancing the lift of the spring. This
backpressure correction factor uses the gas correction factor for choked flow and the liquid
correction factor for non-choked flows.
Comparison of Model Predictions
Darby et al. [21] compared most of the methods discussed herein for predicting the required
relief mass flux for several fairly severe cases involving flashing and (retrograde) condensing
ethylene at several different conditions. They found that most of the equilibrium models and the
HNE model for nozzle lengths greater than 6 inches gave results that were up to 200% higher or
lower than the HDI model, depending upon the value of the relief pressure relative to the
saturation pressure, for conditions well away from the critical point. However, in the vicinity of
the critical point the results varied by up to 6-700%, depending upon how close the relief
pressure is to the saturation pressure (i.e. the degree of sub-cooling). These differences illustrate
that the variation in the results that can arise from applying different models to the same case can
Published in Chemical Enginering, 112, no. 9, pp 42-50, Sept, (2005)
Page 22 15 June 2005
be quite significant, although the trends shown here may not be typical of all conditions that may
arise. Specifically, the Omega, API and HNE methods are not recommended in the vicinity of
the critical point, but may give excellent results under other less stringent conditions, notably for
single component simple fluids far from the critical point, over a small moderate pressure range.
Darby [22] compared the predictions of the HDI method with frozen air/water data
(Lenzing, et al. [24-27]) in four different valves over a range of pressures (see Figs. 1 – 9), and
the HNDI method for steam/water flashing data in one (Leser) valve (Lensing [24, 25]) over a
range of pressures (Figs. 10 – 13). The specifications of the valves are given in Table 1, and the
flow conditions of the tests are shown in Table 2. The manufacturer’s gas flow coefficient, dG
K,
was used in all cases when the flow was choked, and the reported liquid coefficients, dL
K, were
used when the flow was not choked. Note that non-choked flow occurred only when the entering
quality (x) of the mixture was less than 0.001. For the flashing flows, all data points
corresponded to choked flow, and the HNDI method was used with an equilibrium relaxation
length e
L of 40 mm. It should be noted that using different values of the discharge coefficient
for choked and non-choked flow results in a discontinuity at the point corresponding to the
transition point between the two (see e.g. Figure 1). This is realistic, since the actual flow
resistance in choked flow is due only to the nozzle, and is hence lower than that for non-choked
flow where the valve body resistance is also important. The discontinuity is not apparent in the
other Figures, since there are no data points in the immediate vicinity of the choke/non-choke
transition point.
Inlet and Discharge Line Sizing
It is necessary to size the inlet line from the vessel to the relief valve large enough that the
irreversible friction loss in this line is less than 3% of the valve gauge set pressure. This “3%
Published in Chemical Enginering, 112, no. 9, pp 42-50, Sept, (2005)
Page 23 15 June 2005
rule” is specified by API 520 [2] in order to avoid a condition that results in rapid opening and
closing of the valve (“chatter”), with potential damaging consequences. Although the basic
nozzle equations are written in terms of the pressures just upstream and downstream of the valve
nozzle, it is common practice to use the pressure in the protected equipment (stagnation
pressure, o
P ) as the valve inlet (upstream) pressure and the backpressure on the valve
()
B
Pas the
downstream pressure. This ignores the pressure drop in the piping from the vessel to the valve.
This assumption does not introduce a serious error when the inlet pressure drop is low compared
to the set pressure, i.e. when the “3% rule” is satisfied.
Similarly, the irreversible friction loss in the discharge piping should be kept to less than
10% of the valve gauge set pressure, to avoid excessive built-up backpressure which can also
adversely affect the chatter characteristics of the valve. This guideline applies to normal spring
loaded relief valves, but different guidelines apply to balanced bellows and pilot operated valves
(see API [2]).
Recommendations
The HDI/HNDI method is recommended as the calculation method of choice, for both single
phase (gas or liquid) and two-phase flows. It is not subject to the many assumptions/restrictions
that are inherent in the various other methods/models. These restrictions can be very limiting
under certain circumstances, and the identification of these circumstances is difficult to
determine rigorously. The HDI/HNDI method is not only more rigorous but simpler to apply
than the other methods. Its only limitation is the availability of a thermodynamic data base or
model which enables determining the two-phase mixture density as a function of pressure.
Published in Chemical Enginering, 112, no. 9, pp 42-50, Sept, (2005)
Page 24 15 June 2005
Example
To illustrate the procedure for sizing a pressure relief valve, the calculations required to size a
valve for a vessel initially containing saturated water and a small fraction of vapor, will be
shown.
The initial conditions in the vessel are: o
P = 116 psia (8 bar), o
x = 0.001 (quality, or mass
fraction vapor). From steam tables, the entropy of this initial mixture is 0.48946 BTU/lbm. The
valve discharge pressure nb
PP=is 14.64 psia.
The postulated relief scenario requires that a valve be sized to relieve the mixture in the vessel at
a maximum rate of 10,042 kg/hr. A Leser valve will be used, with a dG
K coefficient of 0.77 and
a nozzle length of 40 mm.
The procedure using a spreadsheet is as follows:
1. Starting at the vessel conditions ( o
P =116 psia, o
x =0.001, s = 0.48946 BTU/lbm),
decrease the pressure by 1 psi increments at constant entropy, and obtain the density of
the two-phase mixture m
ρfrom steam tables at each increment. Some steam tables
tabulate this mixture directly; others will give the quality (x) and separate densities of the
liquid and vapor phases, in which case Eqns (7 and 8) can be used to calculate the two-
phase density (setting S = 1).
2. For each pressure increment i, calculate
(
)
i1 i i
PP/
+
−
ρ, where i
ρ
is the average two-phase
density over the increment.
3. Evaluate the summation in Eqn (9) at each interval from o
P to i1
P+(i.e. summing the
terms from step 2 from the initial state to pressure i1
P
+
)
4. Inserting the summation from Step 3 into Eqn (9), and multiplying by the discharge
coefficient dG
K gives the nozzle mass flux n
G corresponding to each exit pressure
(
)
i1
P+.
5. The procedure is repeated at successively lower values of pressure until either (a) the
discharge pressure
b
P is reached, or (b) the mass flux n
G goes through a maximum. If the
maximum is reached before the discharge pressure is reached, this means that the flow is
choked at the nozzle pressure corresponding to the point where the mass flux is
maximum. If no maximum is reached before the discharge pressure is reached, then the
flow is not choked.
This procedure is proper for equilibrium flows, e.g. for nozzles less than 10 cm long. However,
in our example the nozzle length is only 40 mm (4 cm), so non-equilibrium would be expected.
This is easy to account for, as follows:
At each pressure increment, using the values of quality
(
)
e
x, the densities of the liquid
and vapor from the steam tables, and the initial entering quality
(
)
o
x, calculate the equivalent
non-equilibrium quality x from Eqn (27) (using L = 4 cm). This non-equilibrium quality is used
Published in Chemical Enginering, 112, no. 9, pp 42-50, Sept, (2005)
Page 25 15 June 2005
in Eqns ( 7 and 8) to compute the corresponding non-equilibrium two-phase density. This is then
used to evaluate the summation in Eqn (9) and n
G as above.
The results are shown in the attached spreadsheet output. (Note: nodG
GGK/0.9= in the
table. The 0.9 is the recommended “safety factor”, and is incorporated into the value of n
G). It
is noted that assuming equilibrium flow, the calculated mass flux is 4,548 2
kg / sm with choking
at a nozzle pressure of 104 psia, whereas the non-equilibrium flow results in a mass flux of 6,714
2
kg / s m and choking at a nozzle pressure of 96 psia.
In order to select the proper size valve (and nozzle), the required orifice area is calculated
from the computed mass flux and the required relief rate, as follows:
()
2
2
n2
n
m 10, 042 kg / hr 100 cm / m
A 0.644 in
G 2.54in/cm
6, 714kg / s m 3600 s / hr
⎛⎞
== =
⎜⎟
⎝⎠
(Note: this value includes the API “safety factor” of 0.9). This value falls between the area of a
G orifice (0.5674 2
in ) and a H orifice (0.8874 2
in ), so the H orifice would be selected. This
means that the actual flow rate through the nozzle would be:
()
actual
0.8874
m 10,042 kg / hr 13,840 kg / hr
0.644
==
which is the design flow rate that should be used for sizing the inlet and discharge lines.
Published in Chemical Enginering, 112, no. 9, pp 42-50, Sept, (2005)
Page 26 15 June 2005
P T xe RhoM RhoL RhoG RhoAvg DP/RhoM SumDP/rho Gn
psia F lbm/ft^3 lbm/ft^3 lbm/ft^3 lbm/ft^3 ft^2/s^2 ft^2/s^2 kg/sm^2
s = 0.48946
115.9925 338.77 0.001 46.109 56.002 0.2597
115 338.13 0.001756 40.596 56.024 0.25759 43.3525 -106.068 -106.068 2475.302
114 337.48 0.002521 36.149 56.047 0.25547 38.3725 -120.739 -226.807 3223.127
112 336.16 0.004062 29.479 56.094 0.25121 32.814 -282.383 -509.19 3938.271
110 334.83 0.005617 24.717 56.141 0.24696 27.098 -341.948 -851.138 4269.222
108 333.48 0.007188 21.147 56.188 0.2427 22.932 -404.069 -1255.21 4435.672
106 332.11 0.008775 18.373 56.236 0.23843 19.76 -468.933 -1724.14 4516.68
104 330.71 0.010378 16.155 56.285 0.23417 17.264 -536.73 -2260.87 4547.762
102 329.3 0.011998 14.324 56.334 0.2299 15.2395 -608.033 -2868.9 4542.294
100 327.87 0.013635 12.833 56.384 0.22563 13.5785 -682.411 -3551.31 4527.68
98 326.41 0.01529 11.558 56.434 0.22135 12.1955 -759.798 -4311.11 4492.936
96 324.92 0.016963 10.466 56.485 0.21708 11.012 -841.456 -5152.57 4447.803
94 323.42 0.018656 9.5211 56.537 0.2128 9.99355 -927.209 -6079.78 4395.253
92 321.88 0.020369 8.6956 56.589 0.20851 9.10835 -1017.32 -7097.1 4337.035
90 320.32 0.022103 7.9686 56.642 0.20422 8.3321 -1112.1 -8209.19 4274.5
NON-EQUILIBRIUM L = 40mm
x alpha RhoTPa DP/RhoM SumDP/rho Gn
ft^2/s^2
0.001 0.177535 46.1058
0.001302 0.220934 43.70329 -105.2 -105.2 2653.844
0.001608 0.261118 41.47881 -111.68 -216.88 3616.503
0.002225 0.332376 37.53317 -246.84 -463.719 4785.153
0.002847 0.393581 34.14218 -271.356 -735.075 5480.372
0.003475 0.446712 31.19659 -296.977 -1032.05 5933.499
0.00411 0.493254 28.61499 -323.77 -1355.82 6238.037
0.004751 0.534331 26.3353 -351.797 -1707.62 6442.979
0.005399 0.57085 24.307 -381.152 -2088.77 6577.028
0.006054 0.603503 22.49228 -411.904 -2500.68 6659.095
0.006716 0.632872 20.85858 -444.166 -2944.84 6701.449
0.007385 0.659395 19.38221 -477.999 -3422.84 6713.509
0.008062 0.683488 18.04007 -513.561 -3936.4 6701.021
0.008748 0.705451 16.81531 -550.966 -4487.37 6668.893
0.009441 0.725542 15.69401 -590.332 -5077.7 6620.949
Published in Chemical Enginering, 112, no. 9, pp 42-50, Sept, (2005)
Page 27 15 June 2005
NOTATION
A cross sectional area of the nozzle throat (orifice) in a valve, or open area of rupture disk,
()
2
in or
()
2
mm
o
G theoretical mass flux through an isentropic nozzle,
(
)
2
m
lb / s in or
(
)
2
m
lb / hr ft or
()
2
kg / hr cm , etc.
n
G actual mass flux through nozzle = do
KG,
(
)
2
m
lb / s in or
(
)
2
m
lb / hr ft or
()
2
kg / hr cm , etc.
k isentropic exponent for a gas, equal to
p
v
c/cfor ideal gas, (-)
d
K relief valve discharge coefficient (-)
dG
K gas phase discharge coefficient (-)
dL
K liquid phase discharge coefficient (-)
v
K viscosity correction factor for viscous fluids, (-)
L nozzle length, 9in) or (cm), etc.
e
L relaxation length for non-equilibrium flow = 10 cm.
m
required relief mass flow rate,
(
)
m
lb / s or
(
)
m
lb / hr or
(
)
kg / hr , etc.
Re
N Reynolds number through the valve nozzle, using volumetric weighted fluid properties for
mixtures.
P pressure, (psia) or (Pa). etc.
i
P pressure at interval i, (psia) or (Pa). etc.
o
P pressure at valve entrance, (psia) or (Pa). etc.
n
P pressure at the nozzle throat (exit), (psia) or (Pa). etc.
s specific entropy (BTU/ m
lb ) or
(
)
Nm / kg
S slip ratio (ratio of the gas phase velocity to the liquid phase velocity) (-)
x quality, or mass fraction of gas phase, (-)
o
x quality at nozzle entrance, (-)
e
x equilibrium quality at pressure P, (-)
α volume fraction fo the gas phase, (-)
β d/D, ratio of the nozzle diameter to the valve inlet diameter, (-)
ρ density of the fluid (mixture) in the nozzle at pressure P,
(
)
3
m
lb / ft or
()
3
kg / m
G
ρ gas phase density,
()
3
m
lb / ft or
(
)
3
kg / m
L
ρ liquid phase density,
()
3
m
lb / ft or
(
)
3
kg / m
n
ρ fluid density at the nozzle throat at pressure n
P,
(
)
3
m
lb / ft or
(
)
3
kg / m
i
ρ average fluid density over interval i to i+1, (psia) or (Pa). etc.
Published in Chemical Enginering, 112, no. 9, pp 42-50, Sept, (2005)
Page 28 15 June 2005
REFERENCES
1. Wong, W.Y., “Protect Plants Against Overpressure”,, Chemical Engineering, pp. 66 –
73, June, 2001
2. American Petroleum Institute, “Sizing, Selection, and Installation of Pressure-Relieving
Devices in Refineries”, ANSI/API RP 520, 7th Edition, Part 1 Sizing and Selection,
Washington, D.C., January 2000
3. American Petroleum Institute, “Sizing, Selection, and Installation of Pressure-Relieving
Devices in Refineries”, ANSI/API RP 520, 5th Edition, Part 2 Installation, Washington,
D.C., January 2003
4. Center for Chemical Process Safety, “Guidelines for Pressure Relief and Effluent
Handling Systems”, American Institute of Chemical Engineers, New York, NY, 1998
5. The Design Institute for Emergency Relief Systems (DIERS) Project Manual,
“Emergency Relief System Design Using DIERS Tecnology”, American Institute of
Chemical Engineers, 1992
6. R. Darby, AExperiments for Runaway Reactions and Vent Sizing@, Center for
Chemical Process Safety, SACHE Program, AIChE, (ISBN 0-8169-0856-7), November
2001
7. National Fire Protection Association, NFPA-30, “Flammable and Combustible Liquids
Code”, Fullment Center, 11 Tracy Drive, Avon, MA 02322-9910, 2000
8. American Petroleum Institute, “Guide for Pressure Relieving and Depressuring
Systems”, ANSI/API RP 521, Washington, D.C., 2005
9. Das, D.K., “Emergency Relief System Design: In Case of fire, Break Assumptions”,
Chemical Engineering, pp 34 – 40, February, 2004
10. Darby, R., “Chemical Engineering Fluid Mechanics”, 2nd Ed.., Marcel Dekker, NY,
2001
11. Darby, R. and K. Molavi, “Viscosity Correction Factor for Emergency Relief Valves”,
Process Safety Progress, v. 16, no.2, pp 80-82, Summer, 1997
12. Shackleford, A., “Using the Ideal Gas Specific Heat Ratio for Relief-Valve Sizing”,
Chemical Engineering, pp. 54-59, November, 2003
13. Jamerson, S.C. and H.G. Fisher, “Using Constant Slip Ratios to Model Non-Flashing
(Frozen) Two-Phase Flow through Nozzles”, Process Safety Progress, 18 (2), Summer,
1999
14. Darby R, P.R. Meiler, and J. R. Stockton, “Flashing Flow in Nozzles, Pipes and
Valves”, Chemical Engineering Progress, pp 56-64, May, 2001
15. Henry, R E and H K Fauske, “The Two-Phase Critical of One-Component Mixtures in
Nozzles, orifices, and Short Tubes”, Trans. American Society of Mechanical Engineers
J., Heat Transfer, pp 179-187, May , 1971
16. Leung, J. C., “Easily Size Relief Devices and Piping for Two-Phase Flow”, Chemical
Engineering Progress, pp 28-50, 1996
17. Leung, J. C., “The Omega Method for Discharge Rate Evaluation”, pages 367 – 393, in
International Symposium on Runaway Reactions and Pressure Relief Design, editors
G.A.Melhem and H.G. Fisher, American Institute of Chemical Engineers, New York,
NY 1995
Published in Chemical Enginering, 112, no. 9, pp 42-50, Sept, (2005)
Page 29 15 June 2005
18. Simpson, L. L., “Estimate Two-Phase Flow in Safety Devices”, Chem. Eng., pp 98-
102, 1991
19. Simpson, L. L.. “Navigating the Two-Phase Maze”, pages 394 – 415, in International
Symposium on Runaway Reactions and Pressure Relief Design, editors G.A.Melhem
and H.G. Fisher, American Institute of Chemical Engineers Press, 1995.
20. Fauske, H K, “Determine Two-Phase Flows During Releases”, Chemical Engineering
Progress, pp 55-58, February, 1999
21. Darby, R., F.E. Self and V.H. Edwards, “Properly Size Pressure-Relief Valves for Two-
Phase Flow”, Chemical Engineering, pp. 68 – 74, June, 2002
22. Darby, R., “On Two-Phase Frozen and Flashing Flows in Safety Relief Valves.
Recommended Calculation Method and the Proper Use of the Discharge Coefficient”,
J. of Loss Prevention in the Process Industries, 17, 255 -5259, 2004
23. American Petroleum Institute, “Flanged Steel Safety Relief Valves”, API Standard 526,
3rd Edi., Washington, DC, February 1984
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Relief Valve Two-Phase Flow Capacity”, ISO TC 185 WG1 Meeting, Paper N106,
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Published in Chemical Enginering, 112, no. 9, pp 42-50, Sept, (2005)
Page 30 15 June 2005
TABLE I
VALVE SPECIFICATIONS
(Lenzing, et al, [18, 20]
Valve dG
K dL
K Orifice Dia.
(mm)
Orifice Area
()
2
in
B&R DN25/40
(Bopp &
Reuther Si63)
0.86 0.66 20 0.4869
ARI DN25/40
(Albert Richter
901/902)
0.81 0.59 22.5 0.6163
Crosby 1 x 2
“E” (JLT/JBS)
0.962 0.729 13.5 0.2219
Leser DN25/40
(441)
0.77 0.51 23 0.6440
TABLE II
FLOW CONDITIONS
(Lenzing, et al, [18, 20]
Fluid Nom. Pressure
(bar) o
P
(psia)
b
P
(psia)
Air/Water 5 72.495 14.644
Air/Water 8 115.993 14.644
Air/Water 10 144.991 14.644
Steam/Water 5.4 78.295 14.644
Steam/Water 6.8 98.594 14.644
Steam/Water 8 115.993 14.644
Steam/Water 10.6 153.690 14.644
Published in Chemical Enginering, 112, no. 9, pp 42-50, Sept, (2005)
Page 31 15 June 2005
Figure 1 Figure 2
Figure 3 Figure 4
Figure 5 Figure 6
Leser DN25/40, Air/Water, 5 bar
0
5000
10000
15000
20000
25000
0.0001 0.001 0.01 0.1 1
xo
KdGo (kg/sm^2)
Calc
Data
Not
Choked Choked
Crosby 1x2E, Air/Water, 5 bar
0
5000
10000
15000
20000
25000
0.0001 0.001 0.01 0.1 1
Xo
KdGo (kg/sm^2)
Calc
Data
Not Choked
ARI DN25/40, Air/Water, 5 bar
0
4000
8000
12000
16000
20000
0.0001 0.001 0.01 0.1 1
Xo
KdG (kg/sm^2)
Calc
Data
Not choked Choked
B&R DN 25/40, Air/Water, 5 bar
0
5000
10000
15000
20000
25000
0.0001 0.001 0.01 0.1 1
xo
KdGo (kg/sm^2)
Data
Calc
Not Choked
Choked
ARI DN 25/40, Air/Water, 8 bar
0
5000
10000
15000
20000
25000
30000
0.0001 0.001 0.01 0.1 1
xo
KdGo (kg/sm^2)
Data
Calc
Choked
Not Choked
B&R DN 25/40, Air/Water, 8 bar
0
5000
10000
15000
20000
25000
30000
0.0001 0.001 0.01 0.1 1
xo
Data
Calc
not
choked
Choked
Published in Chemical Enginering, 112, no. 9, pp 42-50, Sept, (2005)
Page 32 15 June 2005
Fig. 5
Figure 7 Figure 8
Figure 9
Leser DN 25/40, Air/Water, 8 bar
0
4000
8000
12000
16000
0.001 0.01 0.1 1
xo
KdGo (kg/sm^2)
Data
Calc
all choked
B&R DN25/40 10Bar
Friedel Air/Water Data
0
5000
10000
15000
20000
25000
30000
35000
0.0001 0.001 0.01 0.1 1
xo
KdGo (kg/sm^2)
Data
Calc
Not Choked
Choked
Leser DN25/40, Air/Water 10 Bar
0
5000
10000
15000
20000
25000
0.0001 0.001 0.01 0.1 1
xo
Calc
Data
Not
Choked
Choked
Published in Chemical Enginering, 112, no. 9, pp 42-50, Sept, (2005)
Page 33 15 June 2005
Fig. 10 Fig. 11
Fig. 12 Fig. 13
Leser DN25/40 Valve, Steam/Water, 5.4 bar
0
500
10 0 0
150 0
2000
2500
3000
3500
4000
4500
5000
0.001 0.0 1 0.1 1
xo
KdGo(Kg/sm^2)
Data
HDI
HNDI
L=40
Leser Valve DN25/40, Steam/Water, 6.8 bar
0
1000
2000
3000
4000
5000
6000
7000
0.001 0.01 0.1 1
xo
KdGo(kg/sm^2)
Data
HDI
HNDI
L=40
Leser Valve DN25/40, Steam/Wate r, 8 bar
0
10 0 0
2000
3000
4000
500 0
6000
700 0
8000
0.001 0.01 0.1 1
xo
KfGo (kg/sm^2)
Data
HDI
HNDI
L=40m
m
Leser Valve DN25/40, Steam/Water, 8 bar
0
1000
2000
3000
4000
5000
6000
7000
8000
0.001 0.01 0.1 1
xo
KfGo (kg/sm^2)
Data
HDI
HNDI
L=40m
m
