Sizing of Throttling Device for

Gas/Liquid Two-Phase Flow

Part 2: Control Valves, Orifices,
and Nozzles
Ralf Dienera and Jürgen Schmidtb
a
BASF AG, Inorganic Chemicals Europe, Ludwigshafen, Germany
b
BASF AG, Safety Engineering, Ludwigshafen, Germany; juergen.schmidt@onlinehome.de (for correspondence)

Published online 19 January 2005 in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/prs.10035

The calculation of the mass flow rate through throt- IEC 60534 [1, 2]. The current standards are sufficiently
tling devices is difficult when handling two-phase flow, accurate for the sizing of these devices for single-phase
especially when boiling liquids flow into these fittings. flow, although they do not contain reliable recommen-
Control valves and orifices are typically oversized in dations for two-phase mixtures composed of vapor and
industry and the control range of those valves often liquid. At present there is no appropriate standard ei-
does not fit the control requirements. In this paper the ther nationally or internationally.
HNE-DS method is proposed for the sizing of control In the chemical and petrochemical industries and
valves, orifices, and nozzles in two-phase flow. It ex- also in power plants and offshore facilities, however,
tends the ␻-method, originally developed by Leung, by such a standard is frequently needed. In these plants
adding a boiling delay coefficient to include the degree liquids are often pumped from tanks or pipeline net-
of thermodynamic nonequilibrium at the start of the works into parts of the plants having relatively low
nucleation of small vapor mass fractions upstream of pressures. Under this circumstance the feed rate is
the fitting. The additional introduction of a slip correc- controlled by means of a control valve (see Figure 1). A
tion factor, to take account of hydrodynamic nonequi-
safety valve must as a general rule be installed to
librium (slip), also makes it possible to calculate reli-
protect the plant against overpressure, for instance in
ably the flow rate through control valves and orifices in
both flashing and nonflashing flow. the event that the outlets have become blocked or have
In Part 2 the HNE-DS method for short nozzles, been inadvertently closed. The size of this safety device
orifices, and control valves is considered. Part 1 de- is based on the maximum feed rate through the control
scribes the sizing of safety valves using the same valve—the control valve is part of the safety concept.
method. Additionally, the derivation of the HNE-DS This complicates any replacement of the control valve.
model is given there in detail. The predictive accuracy Therefore, the flow rate is in practice usually limited by
of the HNE-DS model has been checked with reference an additional orifice, which is fitted downstream from
to more than 1300 sets of experimental data. © 2005 the control valve. The control valve is then no longer
American Institute of Chemical Engineers Process Saf relevant to safety and it can be replaced by any other
Prog 24: 29 –37, 2005 valve without having an effect on the safety concept.
Overall, the design engineer is confronted with the task
of estimating the flow rate through the control valve or
INTRODUCTION orifice as accurately as possible to determine the size of
The sizing of control valves, orifices, and nozzles for the safety valve on the low-pressure equipment (Figure
the flow of gases, noncondensing vapor and nonvapor-
1). The sizing task is divided into two steps:
izing liquids is described in the standards ISO 5167 and
1. Sizing a relief valve for two-phase flow (Part 1, see
© 2005 American Institute of Chemical Engineers Literature Cited [19])

Process Safety Progress (Vol.24, No.1) March 2005 29

 taken into consideration. In general, this method may
lead to highly underestimated or distinctly overesti-
mated mass flow rates. Sheldon and Shuder [3] have
recommended a variable correction factor to obtain
less uncertain results. Nevertheless, in the case of boil-
ing liquids or mixtures of flashing liquids and vapors
the results are still unsatisfactory. Diener [4] developed
a more sophisticated physical model including hydro-
dynamic and thermodynamic nonequilibrium flow. It-
eration procedures and a numerical integration of the
equations are necessary for calculating the mass flow
rate. Additionally, pressure- and temperature-depen-
dent property data are needed (density, enthalpy, and
entropy), which in practice are often not available.
The most common method for calculating the mass
Figure 1. Typical layout of production vessel fed by a
flow rate in the case of a gas/liquid two-phase flow for
control valve with a safety valve on top to avoid an technical purposes is the ␻-method, originally devel-
inadmissible vessel overpressure. The size of the
oped by Leung [5, 6]. It is based on the homogeneous
safety valve is determined by the maximum feed
flow model in which gas and liquid flow at the same
through the control valve.
velocity and are uniformly distributed over the flow
cross section. Both phases are in hydrodynamic equi-
librium and the phase boundary is (theoretically) of
infinite size. Under real conditions, however, these
2. Sizing a control valve or orifice for two-phase flow
assumptions are true only for the limiting case in spray
(Part 2)
or wet vapor flow having just a few drops of liquid in
Both steps are based on the same method: the HNE-DS the vapor. Nevertheless, the model is used for the
(homogeneous nonequilibrium method developed by lower vapor content range and in flow without a phase
the authors Diener and Schmidt). This is an extended transition, such as a mixture of air and water. It still
␻-method, originally developed by Leung. In this arti- provides acceptable results for the practitioner even in
cle the HNE-DS method is described for sizing control the median range of vapor contents. The reason for this
valves, orifices, and nozzles. is the acceleration in the valve that mixes phases thor-
oughly and as a result they are largely homogeneously
LITERATURE REVIEW distributed.
A few methods are described in the literature for
two-phase flow with sufficient accuracy for the deter-
mination of the mass flow rate through throttling de- UNCERTAINTIES OF THE ␻-METHOD
vices. These computational methods, however, are In two-phase flow the real mass flow rate can be
very resource intensive. The calculation requires pre- determined only by experiments, for example mea-
cise knowledge of the internal geometry of the fitting, surements made at the Technical University of Ham-
such as the contour of a throttle cone in a control valve, burg-Harburg, in the department of Prof. Dr. L. Friedel,
not usually given by the manufacturer. In addition, and by SAMSON AG (Frankfurt, Germany). There, the
physical properties data such as densities, viscosities, mass flow rate was measured when mixtures of steam
enthalpies, and entropies of the vapor and liquid and boiling water were passed through control valves.
phases as a function of pressure and temperature over The test valves used had nominal diameters of 25, 50,
a broad parametric range are needed. In many cases and 80 mm and had different types of valve cones
these are not available or they must be determined by (V-port cone, parabolic cone, and perforated cone).
means of costly measurement procedures. The test setups and the measuring methods and tech-
Apart from these complex methods there are also nology employed are described by Diener [4]. In Figure
sizing methods, which are easier to use and in line with 2 the mass flow rates calculated by the ␻-method of
actual practice in the chemical and petrochemical in- Leung (homogeneous equilibrium model) are plotted
dustries, but these exhibit certain limits in application. against the measured mass flow rates. If these data sets
Most of the control valves are sized by means of the IEC were in agreement with each other all points would lie
60534-2-1 standard [2]. Any limitation in mass flow on the diagonal. The deviations, however, are far re-
capacity attributed to cavitation or flashing of a dis- moved from the diagonal line and are distributed asym-
tinctly subcooled liquid is taken into account by em- metrically about the 100% error limit. The maximum
pirically developed correction factors. If two-phase deviation between measured and calculated values is
mixtures or boiling liquids are to be considered in the 500%; that is, the true flow rate through the control
inlet of the valve, the standard is not applicable. As a valve is about six times larger than the calculated rate.
consequence, in practice the flow coefficients for sin- A downstream safety valve, sized using these calcu-
gle-phase gas and single-phase liquid are added with lated flow rates through the control valve, would be far
the mass flow rate of each phase as a weighting factor. too small and the calculated relief cross section would
This so-called addition model is a frozen flow consid- likewise be about six times smaller than the cross
eration where no momentum, heat, or mass transfer is section needed. The method is, therefore, unsuitable

30 March 2005 Process Safety Progress (Vol.24, No.1)

 recommended for throttling devices where the accel-
erational pressure drop dominates the frictional pres-
sure drop, that is, in typical short nozzles, orifices, and
control valves. If the area ratio of the nozzle or orifice
tends to 1 (small flow contraction), the boiling delay
exponent decreases. Therefore, the exponent a be-
comes 2/5 for safety valves and for long nozzles and
diffusors a value of a ⫽ 0 is recommended (also see the
Appendix in the companion article, Part 1).
In principle, the HNE-DS method can also be ap-
plied to venturies. In the absence of detailed experi-
mental data with low-quality two-phase flow, the ex-
ponent a cannot precisely be specified. From a
theoretical perspective, the value will be expected to
be 2/5 or less.
Figure 2. Accuracy of reproduction of control valve With the aid of the compressibility factor ␻, includ-
mass flow rates by the ␻-method of Leung for vapor/ ing the boiling delay coefficient N, the critical pressure
liquid flow with low vapor content. ratio ␩crit can now be determined more accurately. By
comparing this with the actual pressure ratio in opera-
tion ␩0 (Eq. 2), the flow condition in the narrowest flow
cross section (critical or subcritical pressure ratio) can
for sizing calculations for inlet flow conditions involv- be determined (Eq. 7). The corresponding pressure
ing boiling liquids with only low vapor contents. ratio ␩crit (critical flow) or ␩0 (subcritical flow) is then
used to calculate the expansion coefficient ␺ (Eq. 8).
RECOMMENDATION FOR AN EXTENDED METHOD (HNE-DS) With this coefficient the mass flux in the narrowest flow
cross section of a (adiabatic) throttling device in a
Mass Flux through an Ideal Nozzle frictionless flow ṁid is obtained (Eq. 9).
To take the boiling delay into account the HNE-DS The equation for the compressibility factor N con-
method was developed. It extends the ␻-method by an tains no additional physical properties and iterations
equation for the boiling delay coefficient. All equations are unnecessary. In contrast with the distinctly more
needed for the application of the HNE-DS method are complex nonequilibrium model of Henry and Fauske
given in Tables 1 and 2. The basic idea for this model [8] the mass flux is defined as a continuous function of
was outlined in 1998 [7]. the vapor mass flow quality and there is no need for
In general, the mass flow rate through nozzles, con- derivatives of property data functions (cf. Figure 3). In
trol valves, and orifices may be calculated by this figure the mass flow densities calculated by both
methods for the flow of steam and water through an
Ṁ CV/orif ⫽ ⌫corrACV/orifṁid with ⌫corr ⫽ ␣CV/orif␾S (1) ideal nozzle are plotted as a function of the stagnation
vapor mass flow qualities at inlet pressures of 10, 1, and
where ACV/orif is the valve seat/orifice area of the fitting, 0.1 MPa (100, 10, and 1 bar). The calculated results are
ṁid is the mass flux through an adiabatic, ideal formed almost identical. Equally high accuracy is also obtained
nozzle in case of frictionless flow, ␣CV/orif is the flow using the refrigerant R12 (cf. Figure 4) whose physical
coefficient, ␾ is the slip correction factor, and S is a properties are very different from those of water. The
safety factor. The flow coefficient for nozzles equals the enthalpy of vaporization for R12 is smaller by more
so-called velocity coefficient with values of 0.95 to 1. than a factor of 10 and its heat capacity is lower by a
In Table 1, the calculation procedure for the mass factor of 4. Nevertheless, the results of both computa-
flow rate through an ideal nozzle ṁid is summarized: tional methods are in good agreement, at least in the
the required variables of state and physical property region of low vapor mass flow qualities up to about
data are specified for the (maximum permissible) stag- 10%. Although the new HNE-DS method is consider-
nation condition pin, Tin in front of the fitting, such as ably simpler to apply, the predictive accuracy is similar
in an upstream vessel or pipe network. In the case of a to that from the more complicated method of Henry
very large pressure drop in front of a control valve or and Fauske.
orifice it may be necessary to determine an imaginary
stagnation condition at the inlet—that is, an isentropic Flow Rate through Control Valves and Orifices
flash calculation from the static inlet conditions to a To determine the mass flow rate through a control
fictitious total condition is made. The compressibility valve or orifice, the flow correction factor ⌫corr has to
factor ␻ (Eq. 4) and the critical pressure ratio ␩crit (Eq. be specified. It represents the ratio of the true mass
5) are then determined for a homogeneous flow in flow rate through the throttling device in comparison to
thermodynamic equilibrium (N ⫽ 1). Based on this first a frictionless flow through an adiabatic ideal nozzle.
estimate the boiling delay factor N is calculated using Table 2 is a compilation of the equations for deter-
Eq. 6b with sufficient accuracy so that the compress- mining the mass flow rate through throttling devices
ibility factor ␻ may be corrected taking account of the with high acceleration of the fluid, such as control
boiling delay. By analogy with the method of Henry valves and orifices, on the basis of the (ideal) mass flux
and Fauske [8] a value for the exponent a ⫽ 3/5 is calculated using the relationships in Table 1. The dis-

Process Safety Progress (Vol.24, No.1) March 2005 31

Table 1. Determination of mass flux for frictionless flow through an adiabatic throttling device (such as nozzle,
orifice, control valve, safety valve).

State variables and property data pin, Tin, pout, ⌬hv,in, cpl,in, vg,in, vl,in, ẋin
pout p0 pcrit pVC
Pressure ratios ␩0 ⫽ ⬵ ␩crit ⫽ ␩⫽ (2)
pin pin pin pin
Homogeneous specific volume vin ⫽ ẋinvg,in ⫹ (1 ⫺ ẋin)vl,in (3)
of mixture
Compressibility factor
(equilibrium condition, N ⫽ 1) ␻N⫽1 ⫽
vin
⫹
vin 冉⌬hv,in 冊
ẋinvg,in cpl,inTinpin vg,in ⫺ vl,in 2 (4)

Critical pressure ratio ␩crit ⫽ 0.55 ⫹ 0.217 䡠 ln ␻N⫽1 ⫺ 0.046 䡠 (ln ␻N⫽1)2 ⫹ 0.004 䡠 (5)
(equilibrium condition, N ⫽ 1) (ln ␻N⫽1)3
␻N⫽1 ⱖ 2 ␩crit2 ⫹ 共␻N⫽12 ⫺ 2 䡠 ␻N⫽1兲 䡠 共1 ⫺ ␩crit兲2 ⫹ 2 䡠 ␻N⫽12 䡠 共␩crit兲 ⫹
␻N⫽1 ⱕ 2 2 䡠 ␻N⫽12 䡠 共1 ⫺ ␩crit兲 ⫽ 0

冉 冊
Compressibility factor
(nonequilibrium condition, ẋinvg,in cpl,inTinpin vg,in ⫺ vl,in 2 (6a)
␻⫽ ⫹ N
N ⱕ 1) vin vin ⌬hv,in

冋
N ⫽ ẋin ⫹ cpl,inTinpin 冉
vg,in ⫺ vl,in
⌬hv,in
2 冊 冉 冊册
ln
1 a
␩crit
(6b)

a ⫽ 3/5 orifices, control valves, short nozzles

a ⫽ 2/5 safety valves (see Part 2), control valve (high lift)
a⬵0 long nozzles, orifice with large area ratios
Critical pressure ratio
␻ⱖ2 ␩crit ⫽ 0.55 ⫹ 0.217 䡠 ln ␻ ⫺ 0.046 䡠 (ln ␻)2 ⫹ 0.004 䡠 (ln ␻)3 (7)
␻ⱕ2 ␩crit
2
⫹ (␻2 ⫺ 2␻)(1 ⫺ ␩crit)2 ⫹ 2␻2ln(␩crit) ⫹ 2␻2(1 ⫺ ␩crit) ⫽ 0
Expansion coefficient
• critical ␩0 ⱕ ␩crit f ␩ ⫽ ␩crit
• subcritical ␩0 ⬎ ␩crit f ␩ ⫽ ␩0

冑 冉冊␻ ln
1
␩
⫺ 共␻ ⫺ 1兲共1 ⫺ ␩兲
(8)

冋冉 冊 册
␺⫽
1
␻ ⫺1 ⫹1
␩

Mass flux for isentropic

frictionless flow ṁid ⫽ ␺ 冑
2p in
vin
(9)

charge coefficient for control valves ␣CV is determined ues quite well (Eq. 13). Results of Eq. 13 are almost as
by the value of Kvs (Eq. 11a). The determination of the accurate as calculated values based on the more com-
discharge coefficient for orifices, that is, the so-called plex model of Benedict [11] for contraction coefficients.
contraction coefficient—the area ratio of the vena con- The discharge coefficients have been validated with
tracta and the inlet pipe—is based on the discharge experimental values of BASF for inlet pressures up to
coefficients for the flow of pure vapor and pure liquid. 300 bar.
Idelchik [9] has recommended the following pressure The discharge coefficient for two-phase flow (Eq.
loss coefficient for sharp-edged orifices in fully devel- 15) is specified similar to the procedure specified in
oped turbulent flow (Re ⬎ 104): Schmidt and Westphal [12, 13], that is, weighting the

冋冉 冊 册 冋冉 冊 册
discharge coefficients for vapor and liquid flow by the
2 2 2 2
dpipe 1 mean void fraction ⑀ in the narrowest flow cross section
␨ pipe ⫽ ⫺1 ⫽ ⫺1 (10)
dVC ␣orif,l (Eq. 14). It is not recommended to weight it by the void
fraction at inlet stagnation condition because that may
This loss coefficient can be recalculated into a dis- lead to a significant error. For example, at the expan-
charge coefficient for pure liquid flow. The compress- sion of boiling water from a tank at a pressure of 10 bar
ibility dependency of the discharge coefficient is recal- into the atmosphere, the steam content at inlet stagna-
culated from experiments of Perry [10] performed with tion condition (tank) is zero, whereas the steam con-
pure gas flow. A trigonometric function with the pres- tent at the narrowest flow cross section is greater than
sure ratio between outlet pressure and inlet stagnation 90 Vol %.
pressure describes the wide range of experimental val- In addition to boiling delay the difference in velocity

32 March 2005 Process Safety Progress (Vol.24, No.1)

Table 2. Determination of discharge capacity through control valves and orifices.

Control Valve Orifice

Data from Table 1 ṁid vl,in, vin, ␻, ṁid, ␩crit, ␩0
Geometric data Kvs dorif (orifice diameter), dpipe
1

冑
Discharge ␣orif,l ⫽
␳ref ⫽ 1000 kg/m3, 2
d orif (11b)
coefficient 1 ⫹ 0.707 1 ⫺ 2
d pipe

共␣A兲CV ⫽ Kvs 冑 ␳ ref

2⌬pref
共11a兲
5 3
c ⫽ ⫹ ␣orif,l
8 8
(12)

c ⫹ ␣orif,l c ⫺ ␣orif,l
␣orif,g ⫽ ⫹ cos共␲␩out兲 (13)
2 2
␩0 ⱕ ␩crit f ␩ ⫽ ␩crit ␩0 ⬎ ␩crit f ␩ ⫽ ␩0 (14)
Vg vl,in

冉 冊
ε⫽ ⫽1⫺
Vg ⫹ Vl 1
␻vin ⫺ 1 ⫹ 1
␩
␲ 2
共␣A兲orif ⫽ d orif 关ε␣orif,g ⫹ 共1 ⫺ ε兲␣orif,l兴 (15)
4

冑 冑 冓 冋冉 冊 册再 冋冉 冊 册冎冔
Slip correction 1/6 5/6 ⫺1/2
(two-phase v in v in vg,in vg,in (16)
␾⫽ ⫽ 1 ⫹ ẋin ⫺ 1 1 ⫹ ẋin ⫺1
multiplier, [14]) ve,in vl,in vl,in vl,in
Mass flow rate Ṁout,S⫽1 ⫽ (␣CV/orif ACV/orif) ␾ṁid (17)
S ⫽ safety factor (recommended values 1–1.3)
Mass flow rate to Ṁout ⫽ ṀCV/orif,S⫽1S ⫽ ṀCV/orif,S⫽11.3 (18)
be discharged

Figure 3. Comparison of the Henry/Fauske model with the HNE-DS method for steam/water flow.

between the gaseous and liquid phases (slip)—the so- to deviations of about 30 –50% in mass flow rates. This
called hydrodynamic nonequilibrium—should also be would lead to a mass flow rate larger than that calcu-
taken into consideration (see the derivation in Appen- lated without taking the nonequilibrium into consider-
dix A). For this purpose Simpson et al. [14] specifies the ation and is thus not conservative. As an example, the
two-phase multiplier, which is based on the effective measured discharge capacities through control valves
specific volume by Lottes [15]. The multiplier has been presented in Figure 2 were recalculated using the
validated with a large volume of measured data for HNE-DS model with and without taking the slip cor-
flow through orifices and valves. rection coefficient into account (see Figures 5 and 6).
Although, in comparison with thermodynamic non- Even in the case of no slip correction, the deviations
equilibrium, the effect of hydrodynamic nonequilib- between measured and calculated values are distinctly
rium is relatively moderate, it can nevertheless give rise smaller than those calculated with the original ␻-meth-

Process Safety Progress (Vol.24, No.1) March 2005 33

Figure 4. Comparison of the Henry/Fauske model, the HNE-DS, and the original ␻ method for R12
vapor/liquid flow.

Figure 5. Accuracy of reproduction of control valve

mass flow rates by means of the HNE-DS method
without slip correction in vapor/liquid flow having Figure 6. Accuracy of reproduction of control valve
low vapor content. mass flow rates by means of the HNE-DS method
with slip correction for steam/water flow having low
vapor content.
od: the mean logarithmic deviation is just 32%, whereas
the variance of the logarithmic deviation is now only
38% (definition of statistic numbers; see Appendix C).
This is much better by comparison with the values of conservative estimate, a safety margin of 30% (factor
126 and 138%, respectively, obtained when the orig- of S ⫽ 1.3) is proposed for the determination of the
inal ␻-method is used. However, deviations of up to maximum mass flow rate. In Figure 7 steam/water
100% are still possible. When the slip correction orifice data from Friedrich [16] are presented. Again,
factor ␾ is applied, the accuracy of reproduction is the accuracy of reproduction of the new HNE-DS
again significantly lowered (see Figure 6). The devi- method is excellent.
ations are distributed almost symmetrically about the In contrast to the data of flashing steam/water flow,
diagonal. The mean logarithmic deviation is ⫺5% Figure 8 shows the comparison of calculated and mea-
(that is, the mass flow rate is slightly high). Thus, on sured data for the flow of nonflashing air and water.
average, the calculation approach yields slightly con- The mean logarithmic deviation is only 21% for all of
servative results. At 17% the variance of the logarith- the 723 data points. Figure 8 underlines the quality of
mic deviation is relatively low. With a few exceptions the slip correction factor because the boiling delay
the deviations are less than 30%. To allow for a factor does not come into play.

34 March 2005 Process Safety Progress (Vol.24, No.1)

Figure 7. Comparison of mass flow rate through orifices according to HNE-DS and measured by Friedrich [16].

sets of steam/water experimental data and more than

700 measurements with air/water. Additionally, the
HNE-DS model was proven for the flow through throt-
tling devices with the refrigerant R12, CO2 /CO2 vapor,
and N2 /N2 vapor. In any case, the HNE-DS model
provides excellent results, even at very low mass flow
qualities at the inlet of the throttling device.
NOMENCLATURE
␣orif ⫽ orifice discharge coefficient for two-
phase flow
␣CV ⫽ control valve discharge coefficient
for two-phase flow
␣CV/orif ⫽ discharge coefficient for a valve or an
orifice for two-phase flow (control
valve ␣V ⫽ ␣CV; orifice ␣V ⫽ ␣orifice)
Figure 8. Accuracy of reproduction of control valve
ACV/orif ⫽ seat area of the control valve (ACV/orif
mass flow rates by means of the HNE-DS method
⫽ ACV) or orifice (ACV/orif ⫽ Aorif)
with slip correction for air/water flow having low
a ⫽ boiling delay exponent (see Table 1)
vapor content.
cpl,in ⫽ specific heat capacity of the liquid at
stagnation state
dorif ⫽ orifice diameter
dpipe ⫽ pipe diameter
SUMMARY ⑀ ⫽ void fraction in the narrowest flow
The ␻-method, originally developed by Leung, is cross section of the throttling device
extended by a boiling delay coefficient to take account ␾ ⫽ slip correction factor
of the delayed boiling of a liquid (thermodynamic non- ⌫corr ⫽ flow correction factor
equilibrium) in a depressurizing flow process. The ex- ␩ ⫽ pressure ratio (ratio of real pressure
tension led to the new HNE-DS method, which is just as in the narrowest flow cross section
easy to use as that developed by Leung, and requires and the inlet stagnation pressure)
physical properties only at the stagnation condition. ␩0 ⫽ back pressure ratio at the outlet of
Resource-intensive equations of state and derivations the control valve or orifice (ratio of
of physical property functions are not needed; nor, as back pressure and the inlet stagna-
a rule, are iterations necessary. Only in the case of low tion pressure)
compressibility factors (␻ ⬍ 2) is it advisable to deter- ␩crit ⫽ critical pressure ratio (ratio of critical
mine the critical pressure ratio by means of the implicit pressure in the narrowest flow cross
equation. section and the inlet stagnation pres-
The advantage of the HNE-DS model is that it can be sure)
applied equally to several throttling devices, such as ⌬hv,in ⫽ latent heat of vaporization at stagna-
control valves, orifices, nozzles, and safety valves (Part tion state
1). The overall reproducibility of the HNE-DS model KVS ⫽ liquid discharge factor for fully
has been checked with reference to more than 1300 opened control valve

Process Safety Progress (Vol.24, No.1) March 2005 35

 ␺ ⫽ expansion factor or outflow function
ṁid ⫽ mass flux through an adiabatic fric-
tionless nozzle (ṁcrit ⫽ Ṁid /ACV/orif)
ṁ id,slip
ṁid
⫽ 冑v in
ve,in
f ṁid,slip ⫽ ␾ṁid with ␾ ⫽ 冑 v in
ve,in
(A2)

ṀV/orif ⫽ mass flow rate through a control

valve or orifice, which has to be dis- The slip-corrected specific volume may be developed
charged from the pressurized system by a momentum balance (so-called momentum-spe-
N ⫽ boiling delay factor cific volume) including the slip factor K according to
pin ⫽ stagnation or inlet stagnation pres- [14]
sure (see Figure 1)
pout ⫽ back pressure at the outlet of the
control valve or orifice (the pressure
that exists at the outlet of a throttling
v e,in ⫽ 共ẋinvg,in ⫹ K 共1 ⫺ ẋin兲vl,in兲 ẋin ⫹ 冋 共1 ⫺ ẋin兲
K 册 (A3)

device)
pcrit ⫽ critical pressure at choking condi-
tions
K⫽ 冉 冊vg,in
vl,in
5/6

pVC ⫽ pressure in the narrowest flow cross

Rearranging Eq. A3 leads to a momentum specific vol-
section (fluid-dynamic pressure oc-
ume model based on the specific volume of both
curring in the narrowest flow cross
phases at inlet stagnation condition.
section of the throttling device)
⌬pref ⫽ reference pressure difference (0.1
MPa)
␳ref ⫽ reference density (1000 kg/m3) 冓
v e,in ⫽ vl,in 1 ⫹ ẋin 冋冉 冊 册 vg,in
vl,in
1/6
⫺1 (A4)

冋冉 冊 册冎冔
Sabs ⫽ variance of the absolute deviations
再
5/6
Sln ⫽ variance of the logarithmic devia- vg,in
⫻ 1 ⫹ ẋin ⫺1
tions vl,in
S ⫽ safety factor (recommended value
1–1.3) APPENDIX B: APPLICATION LIMITS OF THE MODEL
Tin ⫽ inlet stagnation temperature (see Fig- The application range of the HNE-DS method is
ure 1) exactly the same as that for the original ␻-method [5, 6].
vg,in ⫽ specific gas volume at inlet stagna- Special emphasis should be given to the following
tion state assumptions:
vin ⫽ mixture-specific volume at inlet stag-
• Validity of the Clausius–Clapeyron equation. It is
nation state
vl,in ⫽ specific liquid volume at inlet stagna-
proven for single-component vapor/liquid sys-
tion state
tems, but also usable for multicomponent vapor/
␻ ⫽ compressibility factor
liquid systems, if the difference of the boiling
ẋin ⫽ inlet stagnation mass flow quality,
point from each component is less the 100° C.
• Vapor phase behaves as an ideal gas. This holds,
that is, the ratio of the gas mass flow
rate to the total mass flow rate of a
if the stagnation pressure is less than or equal to
two-phase mixture at stagnation state
half of the thermodynamic critical pressure of the
X៮ ln ⫽ mean logarithmic deviation
component (pred ⫽ p/pc ⱕ 0.5) and the temper-
Xi,abs ⫽ absolute deviation between experi-
ature is less or equal to 0.9 times the critical
mental and calculated value
temperature (Tred ⫽ T/Tc ⱕ0.9). Otherwise, a real
Xi,ln ⫽ logarithmic deviation between ex-
gas coefficient has to be introduced into the
perimental and calculated value
method.
Yi,calc ⫽ calculated value, e.g., mass flow rate In general, the HNE-DS method is applicable to every
Yi,exp ⫽ experimental value, e.g., mass flow throttling device in industrial processes. The design
rate engineer needs to assume the contraction rate within
the throttling device and the relaxation time for heat
APPENDIX A: DEFINITION OF THE SLIP FACTOR transfer between both phases. In short throttling de-
The ideal mass flow rate of a frictionless homoge- vices, with large depressurization, an exponent a of 3/5
neous flow through an adiabatic nozzle is defined by is recommended as a first estimate, whereas in less-
the HNE-DS model as pronounced nonequilibrium flows a lower value for
the exponent is recommended.

ṁ id ⫽ ␺ 冑 2p in
vin
(A1) APPENDIX C: DEFINITION OF STATISTIC NUMBERS
The average predictive accuracy of the models is
based on the values obtained for the variance of the
whereby ␺ is the expansion coefficient of the fluid. logarithmic deviations between the experimental and
Considering the slip between gas and liquid phase— calculated values (Table C1). Moreover, the mean log-
the velocity ratio of the averaged gas and liquid veloc- arithmic deviation characterizing the average under- or
ities—would lead to an increased mass flow rate be- overprediction of the experimental values is depicted
cause of the increase in density of the flow: for the sake of completeness. The advantages of using

36 March 2005 Process Safety Progress (Vol.24, No.1)

Table C1. Definition of statistical numbers used to characterize the average predictive accuracy of the models.

Statistical Number Definition

Variance of logarithmic deviations Sln ⫽ exp 冑¥ i⫽1

n

n⫺f⫺1
2
X i,ln
⫺1 Xi,ln ⫽ ln
Yi,exp
Yi,calc

Variance of absolute deviations Sabs ⫽ 冑

¥ i⫽1
n

n⫺f⫺1
2
X i,abs
Xi,abs ⫽ Yi,exp ⫺ Yi,calc

冉 冊冘
n
1 Yi,exp
Mean logarithmic deviation X៮ ln ⫽ exp Xi,ln ⫺ 1 Xi,ln ⫽ ln
n i⫽1
Yi,calc

these parameters are already discussed by Govan [17], 11. Benedict, R.P., Fundamentals of pipe flow, Wiley,
Friedel [18], and Diener [4] and showed in the past to New York, 1980.
allow for a balanced description of the merits of each 12. Schmidt, J. and Westphal, F., Praxisbezogenes
correlation. Vorgehen bei der Auslegung von Sicherheitsven-
tilen und deren Abblaseleitungen für die Durch-
LITERATURE CITED strömung mit Dampf/Flüssigkeits-Gemischen—
1. ISO 5167, Measurement of fluid flow by means of Teil 1 (Practical procedure for the sizing of safety
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Berlin, 2003. nik, 69 (1997), No. 6.
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gleichungen für Fluide unter Einbaubedingungen tilen und deren Abblaseleitungen für die Durch-
(IEC 60534-2-1:1998); Ausgabe 2000 – 03. strömung mit Dampf/Flüssigkeits-Gemischen—
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No. 388, 2000. Flow, Coventry, UK, April 19 –20, 1983.
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June 15, 1998. 18. Friedel, L., Kriterien für die Beurteilung der Vorher-
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Process Safety Progress (Vol.24, No.1) March 2005 37