DOI 10.1007/s10010-006-0043-3
ORI GI NAL ARBE I T E N · ORI GI NAL S
Forsch Ingenieurwes (2007) 71: 47–58
Sizing of nozzles, venturis, oriﬁces, control andsafety valves
for initially sub-cooledgas/liquidtwo-phase ﬂow–
The HNE-DSmethod
J. Schmidt
Received: 20 November 2006 / Published online: 12 January 2007
© Springer-Verlag 2007
Abstract Current standards for sizing nozzles, venturis, ori-
ﬁces, control and safety valves are based on different ﬂow
models, ﬂow coefﬁcients and nomenclature. They are gener-
ally valid only for single-phase gas and liquid ﬂow. Common
to all is the concept of one-dimensional nozzle ﬂowin combi-
nation with a correctionfactor (e.g. the discharge coefﬁcient)
tocorrect fornon-idealities ofthe three-dimensional ﬂow.
With the proposed partial non-equilibriumHNE-DS method
an attempt is made to standardize all sizing procedures by
an appropriate nozzle ﬂow model and to enlarge the appli-
cation range of the standards to two-phase ﬂow. The HNE-
DS method, which was ﬁrst developed for saturated and non-
ﬂashing two-phase ﬂow, is extended for initially sub-cooled
liquidsenteringthethrottlingdevice.Toaccountfornon-
equilibrium effects, i.e. superheated liquid due to rapid de-
pressurisation, thenon-equilibriumcoefﬁcientusedinthe
HNE-DS method is adapted to those inlet ﬂow conditions.
Acomparison with experimental data demonstrates the good
accuracy of the model.
Auslegung von D¨ usen, Venturis, Blenden, Stell-
und Sicherheitsventilen f ¨ ur eingangs unterk¨ uhlte
Gas-Fl ¨ ussigkeits-Str¨ omungen nach der HNE-DS
Methode
Zusammenfassung Die derzeitigen Regelwerke zur Ausle-
gung von D¨ usen, Venturies, Blenden, Stell- und Sicherheits-
ventilenbasierenauf verschiedenenStr¨ omungsmodellen,
Durchﬂusskoefﬁzientenund sind mit verschiedenenNomen-
klaturen beschrieben. Sie gelten nur f¨ ur Einphasenstr¨ omung
J. Schmidt (u)
BASF AG, GCT/S-L511,
Ludwigshafen, Germany
e-mail: juergen.schmidt@onlinehome.de
von Gasen und Fl¨ ussigkeiten. Gemeinsam ist den Modellen
indenRegelwerkendieKombinationauseineridealisier-
tenD¨ usenstr¨ omungundeinemKorrekturfaktor(z.B. dem
Ausﬂusskoefﬁzienten), um die Nicht-Idealit¨ aten der dreidi-
mensionalen realen Str¨ omung zu korrigieren. Die neue HNE-
DSMethode,einD¨ usen-Str¨ omungsmodell mitGasenund
Fl¨ ussigkeiten im partiellen Ungleichgewicht, erlaubt es, die
bestehenden Auslegungsempfehlungenindenverschiedenen
Regelwerken zu vereinheitlichen und gleichzeitig auf Zwei-
phasenstr¨ omungen zu erweitern. Die HNE-DS Methode, die
zun¨ achst f¨ ur siedende und nicht-verdampfende Zweiphasen-
str¨ omungenentwickelt wordenist, wirderweitert f¨ uran-
fangs unterk¨ uhlte Fl¨ ussigkeiten imEintritt der Armaturen.
Ungleichgewichtseffekte, beispielsweise die
¨
Uberhitzung
derFl¨ ussigkeitbeischnellemDruckabfall,werdenmitei-
nem erweiterten Ungleichgewichtsfaktor im HNE-DS Mo-
dell ber¨ ucksichtigt. Der Vergleichmit experimentellenDaten
zeigt die gute Genauigkeit des Modells.
List of symbols
Variable Unit Deﬁnition
a – exponent of the non-equilibrium
coefﬁcient N
A m
2
cross-sectional area of the nozzle throat
(seat aera of valve)
C – ﬂow coefﬁcient
C
crit
– ﬂow coefﬁcient at critical pressure
ratio in the nozzle throat
cp
i,0
J/(kg K) speciﬁc liquid heat capacity at inlet
conditions
d m nozzle throat diameter
d
0
m nozzle inlet diameter
K
d,2ph
– derated two-phase ﬂow valve
discharge coefﬁcient
1 3
48 Forsch Ingenieurwes (2007) 71: 47–58
K
d,g
– certiﬁed (derated) valve discharge
coefﬁcient for single-phase
gas/vapor ﬂow
K
d,l
– certiﬁed (derated) valve discharge
coefﬁcient for single-phase liquid
ﬂow
l
Pipe
m length of piping behind the nozzle
throat with an diameter equal to the
nozzle throat diameter
˙ m kg/(m
2
s) mass ﬂux
N – non-equilibriumcoefﬁcient
p Pa pressure in the nozzle throat
p
0
Pa nozzle inlet pressure
p
s
(T
0
) Pa saturation pressure at inlet
temperature
p
b
Pa back pressure
p
c
Pa thermodynamic critical pressure
Q
m
kg/s mass ﬂow rate through the nozzle
T
0
K nozzle inlet temperature
T
c
K thermodynamic critical temperature
v m
3
/kg speciﬁc volume in the nozzle throat
v
0
m
3
/kg speciﬁc volume in the nozzle inlet
v
∗
m
3
/kg dimensionless speciﬁc volume
˙ x
0
– mass ﬂow quality in the nozzle inlet
˙ x
eq
– mass ﬂow quality in the nozzle throat
under thermodynamic equilibrium
conditions
∆˙ x
eq
– change of mass ﬂow quality between
nozzle inlet and throat under
thermodynamic equilibrium
conditions
ε – void fraction in the nozzle throat
β – diameter ratio
η – pressure ratio
η
b
– ratio of back pressure to the inlet
pressure
η
crit
– critical pressure ratio
η
S
– ratio of the saturation pressure
corresponding to the nozzle inlet
temperature (measure of liquid
subcooling) to the inlet pressure
κ – Isentropic coefﬁcient
λ
insul
W/(m
2
K) heat transfer coefﬁcient of the
insulation
ω – compressibility coefﬁcient
ω(N) – compressibility coefﬁcient depending
on the non-equilibriumcoefﬁcient N
ω
eq
– compressibility coefﬁcient for
a homogeneous ﬂuid under
thermodynamic conditions,
ω (N =1)
∆h
v,0
J/kg latent heat of vaporization at inlet
condition
1 Introduction
Two-phasemassﬂowratesthroughthrottlingdevicesare
generally calculated basedonsimpliﬁedgeometries. Most
often a nozzle is considered. The result is then corrected by
anexperimentally determined factor,i.e.africtionordis-
charge coefﬁcient, to account for any deviation in the ﬂow
due to the real geometry – an oriﬁce, a venturi or valve. Cur-
rentlytheonlystandards whichexist areforthesizingof
safety valves given in API 520 [1] and ISO 23521 [2]. These
are based on the world wide acceptedω-method developed
by J.C. Leung, [3, 4], and recommended by the DIERS In-
stitute[5]. Several classical sizingtext booksalsomake
reference toit[6, 7].Oneofthemajoradvantages ofthis
methodisitsuseofknown oreasilymeasurable property
data at inlet stagnation condition of the safety valve.
Theuseoftheω-methodforsaturatedtwo-phaseﬂow
generallyleadstoconservative sizingresults,becauseho-
mogeneousequilibriumﬂowthroughthesafetyvalveis
assumed. However, ifaninitiallysub-cooledoraboiling
liquidwithalowmassﬂowqualityhastobeconsidered
at theinlet of asafetyvalve, it iswell known, that the
method provides an in-acceptable over-estimation of the re-
quired size[e.g.8,9]. Asanalternative, theHenry/Fauske
model [10]whichisbasedonamoreaccurateﬁt toex-
perimental datacanbeappliedinsuchsituationstocal-
culate the mass ﬂowrate. Henry andFauske also pro-
poseda boilingdelayfactor toaccount for the thermal
non-equilibrium of the ﬂuid and get excellent results when
compared with the ﬂow through nozzles. Unfortunately, the
model of Henry/Fauske is based on physical property data
which are only rarely available in industry and additionally
it is more complicate to use than the ω-method.
To overcome the conservatism of theω-method for low
quality inlet ﬂow, both the ω-method and the Henry/Fauske
modelarecombinedintotheHNE-DSmethod(Homoge-
neous Non-Equilibrium Method of the authors Diener and
Schmidt) to account for thermodynamic non-equilibriumef-
fects[11, 12]. Mechanicalnon-equilibriumeffectsareac-
countedfor bymeansof aslipmodel. Inthis way, the
previous work of J.C. Leung and Henry and Fauske is rec-
ognized and engineers in practice may continue to use their
traditional methods, like API 520.
Inadditiontothesizingofsafetyvalves,ageneraliza-
tion of the HNE-DS method is proposed for sizing nozzles,
venturis, oriﬁces, control valves and other throttling devices.
The HNE-DS method is part of the standard ISO/DIS 4126-
10 for safety valves [15] (the standard was accepted in 2006
as a draft international standard, a preliminary standard has
beenpublishedin[13, 14])andisproposedforinclusion
in IEC 60534 (control valves) [16]. Additionally, it is rec-
ommendedtoextendISO5167[17] andISO9300[18]
(nozzles, oriﬁces, venturis) for two-phase ﬂow.
1 3
Forsch Ingenieurwes (2007) 71: 47–58 49
Inthefollowing, theHNE-DSmethodisderivedand
extended for initially sub-cooled liquids at the inlet of throt-
tling devices.
2 HNE-DS method
The basic idea of the HNE-DS method is to consider a throt-
tlingdeviceasafrictionless, adiabaticnozzle. Theﬂuid
is assumed to be a quasi single-phase, i.e. a homogeneous
mixtureofgasandliquidinequilibrium, withtwo-phase
properties.Correctionofthesimpliﬁedmodelaredeﬁned
fornon-idealitieslikeboilingdelayandslipbetweengas
andliquidphase, whichmaybecharacteristicforcertain
throttling devices. Those non-idealities are induced, e.g., by
acontractionandredirectionoftheﬂowandduetofric-
tionandwallheat exchange. The more precise thenozzle
ﬂowmodel accountsfornon-equilibriumeffectsandreal
properties of the ﬂuids, the fewer dependencies have to be
taken into consideration for a discharge coefﬁcient. In any
case,thedischarge coefﬁcient mustbe experimentally de-
termined, at least at certain, representative ﬂow conditions.
A precise nozzle ﬂow model is critical in order to extrapo-
late the ﬂow coefﬁcient of a throttling device from labora-
tory test conditions to ﬂow conditions typically encountered
in industry.
Theone-dimensional momentumbalancefor theﬂow
through a frictionless, adiabatic nozzle with no gravity ef-
fects encountered is,
C =
−
η
η
0
v
∗
dη
(v
∗
)
2
−β
4
; η =
p
p
0
; v
∗
=
v
v
0
; β =
d
d
0
. (1)
The nomenclature of ISO/DIS 4126-10 is identically applied
in the present paper.
Herein, η is the ratio of the pressure in the nozzle throat p
and the inlet p
0
(symbols without subscripts refer to the noz-
zle throat while the subscript ,,0“ stands for the inlet of the
nozzle),v
∗
is the speciﬁc volume ratio andβthe diameter
ratio. The ﬂow coefﬁcient C is, by deﬁnition, the normalized
mass ﬂow rate Q
m
through the nozzle,
C =
Q
m
A·
2·
p
0
v
0
; A =
π
4
d
2
. (2)
Equation 1 is valid for single phase gas and liquid ﬂow as
well asfortwo-phasemixtures. Thegas/liquidtwo-phase
ﬂow is treated as quasi single-phase ﬂow with a speciﬁc vol-
umeofamixture. Anyinformationaboutinterfacialheat
and mass transfer between the phases is included in the di-
mensionlessspeciﬁcvolumeoftheﬂowv
∗
. Itstimeand
cross-sectional average for a homogeneous two-phase ﬂow
under thermodynamic equilibrium condition is [5, pp. 58ff],
v
∗
= ˙ x ·
v
g
v
0
+(1− ˙ x) ·
v
l
v
0
, (3)
from where an equation of statev
∗
(η) for the proﬁle along
the nozzle may be determined by integration of the deriva-
tive,
v
0
dv
∗
dη
=
v
g
−v
l
·
d ˙ x
dη
+ ˙ x ·
dv
g
dη
+(1− ˙ x) ·
dv
l
dη
(4)
Theliquidmaybeconsideredas almost incompressible,
the gas as following the ideal gas law and the boiling line
ofthegas/liquidmixturemaybeprescribedbytheClau-
siusClapeyronslaw.Heatofvaporization andliquidspe-
ciﬁcvolumeareconstant andthetemperatureofthegas
istakenequaltothat oftheliquid(spontaneousheat ex-
change). What remains is the information about the change
of mass ﬂow quality along the pressure curve in the nozzle,
i.e. the interfacial heat and mass transfer. Any heat transfer
limitationduetosteeppressuregradientswithinthenoz-
zlewill result intoa thermodynamic non-equilibrium –or
boilingdelay–oftheﬂow. HenryandFauske[10]iden-
tiﬁedtheboilingdelayas adeviationof themass ﬂow
qualityfromequilibriumconditionsat acertainpressure
drop
d ˙ x
dη
=
d ˙ x
eq
dη
· N . (5)
The proposed dependency of the boiling delay factorNon
the mass ﬂow quality is linear at low qualities and constant
for mass ﬂow qualities larger than 0.14. In contrast, Diener
andSchmidt[11, 12]suggesteda(continuous)power-law
function for the boiling delay factor. The basis of their func-
tion is the mass ﬂow quality in the narrowest cross section
of the nozzle ˙ x
eq
– the nozzle throat – if both the vapor and
liquid phase are in thermal equilibrium:
N =
¸
˙ x
eq
¸
a
; a ∈ 0 . . . ∞; ˙ x
eq
∈ 0 . . . 1 ; ⇒N ∈ 0 . . . 1 .
(6)
Theexponent“a”dependsontherelaxationtimeforthe
two-phase ﬂow in the nozzle up to its narrowest cross sec-
tion. Inaveryshortnozzle, themomentumandheat ex-
change between both phases is poor and, hence, the boiling
delay reaches its maximum (a →∝; N →0). The ﬂow is al-
most frozen. If vapor and liquid would have time to reach its
equilibrium state, i.e. in a very long nozzle, almost no boil-
ing delay will occur (a →0; N →1). As a general rule, the
larger the inlet mass ﬂow quality is, the less pronounced is
the boiling delay effect, Fig. 1.
1 3
50 Forsch Ingenieurwes (2007) 71: 47–58
Fig. 1 Mass ﬂux calculated withthe HEMand the Frozen Flow
model versus inlet mass ﬂowqualityfor anozzle investigatedby
Sozzi/Sutherland [22]
Followingthederivationof theoriginal ω-method[3]
and taking the boiling delay coefﬁcient N into consideration
yields anequation ofstatefor atwo-phase ﬂow including
thermal non-equilibriumeffects,
v
∗
=ω(N)
1
η
−
1
η
0
−1 , (7)
where the compressibility coefﬁcient is deﬁned by,
ω(N) =
1
κ
˙ x
0
· v
g0
v
0
+
cp
l0
· T
0
· p
0
· η
0
v
0
·
¸
v
g0
−v
l0
∆h
v0
¸
2
· N , (8)
N =
˙ x
0
+cp
l0
· T
0
· p
0
· η
0
·
v
g0
−v
l0
∆h
2
v0
· ln
η
0
η
a
. (9)
The factor 1/κ in the left term on the right hand side was in-
troduced to account for an isentropic rather than an isother-
malchange ofstateinsingle phasegas ﬂowcompared to
a two-phase gas liquid ﬂow. The compressibility coefﬁcient
ω(N) leads to the original ω-parameter, when vapor and li-
quid phase are in thermal equilibrium (N =1; ω(N =1) =
ω
eq
). Ifthereisnomassandheattransferbetweenvapor
and liquid at all (frozen ﬂow) or in a non-ﬂashing gas/liquid
ﬂow, the second term on the right hand side of Eq. 8 would
vanish (N =0).
Due to simpliﬁed assumptions, such as constant heat of
vaporization, Eq. 7 results in large unacceptable uncertain-
tiesclosetothethermodynamiccritical point of aﬂuid.
Hence, it shouldonlybeused, if thereducedinlet pres-
sureor temperatureof theﬂuiddonot exceedvaluesof
p
0
/p
c
≤ 0.5 orT
0
/T
c
≤ 0.9, respectively. Additionally, for
multi-component ﬂuids the largest boiling temperature dif-
ference oftwocompounds shouldbelessthan100
◦
C.In
any other case, a more precise equation of state should be
considered [7, 19].
3 Initially sub-cooled liquid ﬂow
If a sub-cooled liquid enters a nozzle three distinct void pro-
ﬁlesmaydevelopuptothenozzlethroat,Fig. 2:(I)pure
liquidﬂowthroughout thenozzleor (II) just ﬂashingin
thenozzlethroat (highlysub-cooledtwo-phaseﬂow)and
(III)ﬂashingpriortonozzlethroat (lowsub-cooledtwo-
phase ﬂow). Letp
b
be the back pressure downstream of the
nozzle throat, then the ﬂow coefﬁcient for pure liquid ﬂow
yields (proﬁle I)
C
l
=
1−η
b
1−β
4
; η
b
=
p
b
p
0
. (10)
Flashing of the liquid in the nozzle throat will cause a local
choke. A ﬁrst estimate for the throat pressure is the satura-
tion pressure of the liquid at inlet temperaturep
S
(T
0
), the
ﬂow coefﬁcient becomes (proﬁle II)
C =
1−η
S
1−β
4
; η
S
=
p
S
(T
0
)
p
0
. (11)
Onlyif gas andliquidphasearehomogeneouslymixed
and in thermodynamic equilibrium can the ﬂow coefﬁcient,
Eq. 11, bereached. Superheatingoftheliquidphasetyp-
icallycauses larger mass ﬂowrates. This effect canbe
accountedfor bya ﬂashingdelay–or moregeneral –
non-equilibrium coefﬁcient Nasalready proposed forthe
HNE-DS model in saturated two-phase ﬂow. According to
Leung[20]theintegralinEq. 1shouldbeseparatedinto
a ﬂow region for liquid ﬂow up toη
S
(s.Eq. 11)and into
a two-phase region starting at η
0
=η
S
until the nozzle throat
toyield agenerally valid ﬂowcoefﬁcient for single-phase
gas and liquid ﬂow as well as for homogeneous two-phase
Fig. 2 Void proﬁles in
initially sub-cooled
two-phase ﬂow through
a nozzle
1 3
Forsch Ingenieurwes (2007) 71: 47–58 51
gas/liquid mixtures,
C =
(1−η
S
) −
η
η
S
v
∗
(ω(N)) · dη
(v
∗
(ω(N)))
2
−β
4
; η
S
=
p
S
(T
0
)
p
0
. (12)
The pressure ratio η, i.e. the pressure in the nozzle throat to
the inlet pressure, which is used in Eq. 7 to deﬁne v
∗
and in
Eq. 9tocalculateN, equal theback pressure ratioη = η
b
incaseofsub-critical ﬂowandthecritical pressureratio
η =η
crit
, if the ﬂow in the nozzle throat is choked. The crit-
ical pressure ratio η
crit
is deﬁned as the ratio where the ﬂow
coefﬁcient C(η), (s. Eqs. 12, 7–9), reachesitsmaximum
value.
Foraplenuminletﬂow(β = 0)andanon-equilibrium
coefﬁcient Nindependentof thepressureratioη, Eq. 12
would lead to an analytical solution for the ﬂow coefﬁcient
C =
(1−η
S
) +
¸
ω(N) · η
S
· ln
η
S
η
−(ω(N) −1) (η
S
−η)
¸
ω(N)
η
S
η
−1
+1
.
(13)
Overall, Eqs. 12, 7–9(integral solution)orEqs. 13, 8, 9
(analyticalsolution)areapplicableforallﬂowconditions
typically encountered in industry.
Flow characteristic
Single phase liquid ˙ x
0
=0;v
∗
=1;η
0
=1;η =η
b
Single phase gas ˙ x
0
=1;v
∗
=v
g
/v
g,0
;
η
0
=1;η ≥η
crit
&η ≥η
b
Initially sub-cooled
two-phase ﬂow
˙ x
0
=0;η
0
=η
S
;η ≥η
crit
&η ≥η
b
Saturated
two-phase ﬂow
˙ x
0
≥0;η
0
=1;η ≥η
crit
&η ≥η
b
Non-ﬂashing
two-phase ﬂow
˙ x = ˙ x
0
=const;
η
0
=1;η ≥η
crit
&η ≥η
b
;N ≡1
4 Critical mass ﬂow rate
The critical mass ﬂow rate is deﬁned as the maximum ﬂow
rate through the nozzle for given inlet conditions,
dC
dη
=0 ⇒max[C(η);η ∈ η
b
, 1] , (14)
and is determined most accurately by integration of Eq. 12
or using Eq. 13 and a subsequent maximum search, Eq. 14.
At its maximum, either the back pressure or the critical pres-
sureisreachedinthenozzlethroat. Typical solutionsof
Eq. 14 for Care presented in Fig. 3 for a constant value of
the exponent a. A sub-cooling ofη
S
=1 represents an ini-
tially saturated liquid and a value ofη
S
=0.5 gives typical
results for initially high sub-cooled liquids, see Eq. 10.
Fig. 3 Flow coefﬁcient according to the HNE-DS model as a function
of pressure ratio for certain degree of initially sub-cooled liquids
AnanalyticalsolutionofEq. 14maybeperformed for
a plenum ﬂow (β =0) of a homogenous mixed ﬂuid in ther-
modynamic equilibrium as proposed by Leung [20]
C
crit,HEM
=
(1−η
S
) +
¸
ω
eq
· η
S
· ln
η
S
η
−(ω
eq
−1)(η
S
−η)
¸
ω
eq
η
S
η
−1
+1
(15)
Herein, ω
eq
is the compressibilitycoefﬁcient for N = 1
(seeEqs.8and9, ω(N =1) =ω
eq
).InEq. 15isη = η
crit
ifη
crit
≤η
S
(low sub-cooling), otherwise η =η
S
(high sub-
cooling) leading to
C
crit,HEM
=
1−η
S
. (16)
The critical pressure ratio η
crit
at the critical ﬂow coefﬁcient
wasderivedbyapplyingEq. 14yieldingatranscendental
equation
0 =
ω
2
eq
−2ω
eq
+1
2ω
eq
η
S
η
2
crit
−2(ω
eq
−1)η
crit
+ω
eq
ln
η
crit
η
S
+
3
2
ω
eq
η
S
−1 . (17)
Equations 15 and 17 are good approximations for two-phase
ﬂow with saturated mixtures (η
s
=1). In case of sub-cooled
liquidstheresultsarepoor, becausethecritical pressure
ratio is highly overestimated. There is a strong dependence
of the ﬂow coefﬁcient on the nozzle length and the degree of
sub-cooling as can be seen from the well known experimen-
tal data of Sozzi and Sutherland [22], Fig. 4. Additionally,
thelowsub-cooledregionislimitedtoveryhighsatura-
tion pressure ratios η
S
, typically in the range of 0.9 to 1, i.e.
tovery smallsub-cooling temperatures, Fig. 5.Atlow in-
let pressures even 1 K makes the difference between the low
sub-cooled region (ﬂashing within the nozzle) and the high
sub-cooled region (ﬂashing in the nozzle throat).
1 3
52 Forsch Ingenieurwes (2007) 71: 47–58
Fig. 4 Uncertaintyof HEMmodel comparedtoexperiments from
Sozzi andSutherland[22]forinitiallysub-cooledsteam/waterﬂow
through nozzles of different length
Fig. 5 Sub-coolingof liquidat nozzleentranceversusnozzleinlet
pressure for experiments given in the open literature
Most of the literature data are measured in the high sub-
coolingregionandtheﬂowratescalculatedbasedonthe
HEM assumption are signiﬁcant too low. As a consequence
of the small region of low sub-cooling, the thermodynamic
equilibrium model ofEq. 15 insteadofEq. 14ismostof-
ten applied in industry in case of initially sub-cooled liquids.
Overall, the HEM model can not be recommended for ini-
tially sub-cooled two-phase ﬂow.
5 Non-equilibrium coefﬁcient N
Intheliteraturethereareatleasttwogeneralmethodsto
account for the thermal non-equilibrium inﬂashingﬂows:
one method is based on the growth of a single bubble fol-
lowing a certain pressure drop [23]. Due to the lack of any
data, the total number of nuclei in a liquid has to be deﬁned
empirically and the models are highly sensitive to this pa-
rameter. Therefore, these types of models can not be applied
for industrial purposes. The second type of model is based
on rapid depressurization experiments due to pipe ruptures
investigatedforthenuclearindustry[24, 25]. Withinmil-
liseconds the pressure falls locally very much below the sat-
uration pressure, and is followed by a vapor explosion. The
depressurization gradient within those experiments is much
larger than in typical throttling devices. Hence, this theory
isalsonot applicableforcalculatingthenon-equilibrium
effects. Overall, there is no physical method available to ac-
count for the thermal non-equilibriumeffects which occur in
nozzles, venturis, oriﬁces or valves.
IntheHNE-DS methodanalternative approach ispro-
posed, basedonasemi-empiricalnon-equilibriumcoefﬁ-
cient N. Physically,thecoefﬁcient Nisameasureofthe
relaxationtimetoexchangeheat andmassbetweenboth
phasesanddependsonthegeometryofthethrottlingde-
vice as well as on the distribution of gas and liquid phase.
It therefore represents both the degree of super-saturation in
the sub-cooled liquid region and boiling delay in saturated
two-phase ﬂow,
N =
¸
˙ x
0
+∆˙ x
eq,nozzle
¸
a
; N ∈ 0 . . . 1 . (18)
The exponent “a” is derived from experimental nozzle ﬂow
data. Thecorrelationwithexperimental measurementsis
an ongoing process, depending on the data available in the
literature.
Diener and Schmidt [11, 12] proposed a value of a =2/5
for safety valves and a =3/5 for control valves for saturated
two-phase gas/liquidﬂowbycomparison withlimitedex-
perimental data from valves. Good agreement for inlet mass
ﬂow qualities larger than 0.05 has been shown. Due to the
lack of more detailed data, no recommendations have been
given for lower mass ﬂow qualities or initiallysub-cooled
two-phase ﬂow.
In an initially sub-cooled two-phase ﬂow the ﬂashing de-
lay or non-equilibrium coefﬁcient Ndepends on the degree
of sub-cooling. This is shown by the comparison in Fig. 6
oftheﬂowcoefﬁcientforhomogeneous equilibriumﬂow
C
crit,HEM
, Eq. 14, andexperimentallydeterminedﬂowco-
efﬁcients C
exp
, Eq. 2. The lower the degree of sub-cooling
is (η
S
→1), thelarger is thedeviationfromHEM. Ex-
perimentallydetermined valuesoftheﬂowcoefﬁcient are
upto5timeslarger, thancalculatedvaluesbasedonthe
HEM model. This is equally true for nozzles and for safety
valves.
The exponent a in Eq. 18 for sub-cooled two-phase ﬂow
( ˙ x
0
≡ 0) was determinedbyaregressionanalysisof lit-
eraturedatameasuredusingnozzlesandsafetyvalvesas
follows
a =
7.5
l
pipe
d
0
+7.5
· (η
S
)
−0.6
. (19)
1 3
Forsch Ingenieurwes (2007) 71: 47–58 53
Fig. 6 Ratio of ﬂow coefﬁcient according to HEM and from measure-
ments on nozzles and safety valves versus sub-cooling
Herein, l
pipe
is the length of pipe with a diameter equal to the
nozzle throat diameter behind the nozzle throat
1
.For noz-
zles without a pipe tail, Eq. 20, reduces to
a =(η
S
)
−0.6
. (20)
Incaseofasaturatedgas/liquidtwo-phaseﬂow( ˙ x
0
≥ 0;
η
0
=1; η ≥η
crit
) Eqs. 13, 8, 9 reduces to the HNE-DS model
already proposed by Diener and Schmidt [11, 12].
6 Validation of the extended HNE-DS model
Literature data fromnozzles, Table 1 [22, 26–29], have been
used for the comparison of the HNE-DS model extended to
initially sub-cooled liquids with experimental results. There
is no unambiguous tendency concerning the nozzle diameter
and length. Additionally, even very small deviations of mass
ﬂow quality result in large deviations of the mass ﬂow rate at
low sub-cooling. Physically, any gas dissolved in the liquid
phase or absorbed on the wall surface of the nozzle may act
as a nucleation source. Hence, the experimental data often
has a high degree of uncertainty.
The HNE-DSmodel extendedto initially sub-cooled
liquids by means of the non-equilibriumcoefﬁcient N
leadstofairlygoodresultsincomparisontoexperimen-
tal data. Figure7showstheoverall tendencydepending
on the degree of sub-cooling compared to the data of
Lee[27]. Especiallyat lowsub-coolingthemodel isby
far better thantheHEMmodel. Afrozenﬂowassump-
tion (no vaporization) would give highly overestimated
results.
Figure8showsthecomparisonoftheHNE-DSmodel
withthe ∼ 1500investigatednozzledatawithsub-cooled
liquidsatthe inlet.Considering theuncertainty oftheex-
1
Nozzles with pipes of up to 500 mm length have been considered in
this study
Fig. 7 Comparisonof ﬂowcoefﬁcients measuredbyLee[27] and
calculated with the HNE-DS model extended for initially sub-cooled
liquids
Fig. 8 Mass ﬂowrateaccordingtotheextendedHNE-DSmethod
and experimentally determined for initially sub-cooled tow-phase ﬂow
through nozzles
perimental datatheHNE-DSmodel isingoodagreement
with the measurements. Of course, it is still possible to fur-
ther improve the correlation, but that would lead to a more
complex function of the exponent a in the model, beside the
fact, that there are not detailed data available about the inﬂu-
ence of the different parameters.
Beside the nozzle data, roughly 2000 measurements
fromsafetyvalves,Table 2[30–33], have alsobeencom-
pared with the HNE-DS model. As proposed by Diener and
Schmidt [11], the valve discharge coefﬁcient for two-phase
ﬂowK
d,2ph
was based on the discharge coefﬁcients for gas
ﬂow K
d,g
and liquid ﬂow K
d,l
, in general given by the valve
manufacturer,
K
d,2ph
=ε · K
d,g
+(1−ε) · K
d,l
. (21)
where εisthevoidfractioninthenarrowest ﬂowcross
section. DienerandSchmidt[11, 12]proposedtousethe
1 3
54 Forsch Ingenieurwes (2007) 71: 47–58
Table 1 Experimental data from nozzles and venturis with initially sub-cooled two-phase depicted from literature
Literature Nozzle Throat diameter Inlet length Outlet length Inlet pressure Inlet temperature ﬂuid
Source Type [mm] [mm] [mm] [bar] [
◦
C]
Sozzi, Sutherland No 1 12.7 44.5 114 54–69 220–285 demineralized
1975 [22] Venuri outlet conus water
No 2 12.7 44.5 0
Rounded Nozzle 12.7
with Tail Pipe 38.1
63.5
190.5
317.5
508
635
1778
No 4 12.7 – 4.7
Sharped edged nozzle 195.2
322
520.7
639
No 5 19 44.5 –
Rounded Nozzle
No 6 54 732 380
Venturi outlet conus
No 6 76.2 696 380
Venturi outlet conus
No 7 28 63.5 165
Venturi outlet conus
Boivin 1979 [26] Nozzle with tail pipe 12 50 450 20–90 200–300 water
30 130 1600
50 130 1700
Veneau 1992 [29] Nozzle with tail pipe 2 2 1.2 280–300 60–120 propane
5 6 3
Lee, Swinnerton Nozzle Sharped 1.8 – 1.8 140–300 200–430 water
1983 [27] Edged Inlet
Nozzle rounded Inlet 1.8 1.3 5.3
Nozzle rounded Inlet 2.5 1.3 7.6
(outlet guide)
Nozzle rounded Inlet 2.5 1.3 7.6
Nozzle rounded Inlet 2.8 1.3 8.4
Simoneau, Hendricks Venturi outlet conus 3.555 74 237 6–140 −244–12 nitrogen
1984 [28] methane
2 D nozzle H¨ ohe: 1.09 205 106
Breite: 10.1
Venturi outlet conus 2.934 7.8 54
Venturi outlet conus 3.555 237 74
D¨ use 2 6.4 k.A. k.A.
D¨ use 3 4 k.A. k.A.
critical pressureratioη = η
crit
inEq. 22underthermody-
namic equilibrium conditions for simplicity.
ε =1−
v
l,0
v
0
·
¸
ω·
1
η
−1
+1
¸, (22)
TheHNE-DSmodel gives excellent agreement withthe
measured valve data even for initially sub-cooled two-phase
ﬂowasshowninFig. 9. Themeanlogarithmicdeviation
–seedeﬁnitionTable 3–isonly16%. TheHEMmodel
recommendedbyAPI 520andISO23521is less accu-
rate.The experiments arehighly under-estimated, Fig. 10,
with a mean logarithmic deviation of 78%. Even the model
ofDarby[35]tends, ingeneral, tounderestimatetheex-
perimental data. Thereasoncouldbethat themodel has
onlybeenﬁttedtoa certainnozzle usedbySozzi and
Sutherland.
An overall comparison of the HNE-DS model with more
than 4000 data including the data with saturated two-phase
ﬂow and non-ﬂashing ﬂow is given in [36].
The HNE-DS model can equally applied to control
valves, oriﬁcesandother throttlingdevices. For initially
sub-cooled liquids the here presented ﬂashing delay factor
1 3
Forsch Ingenieurwes (2007) 71: 47–58 55
Literature Valve Type Seat Diameter Inlet length Outlet length
Source [mm] [mm] [mm]
Boccardi 2005 [31] Leser 10 – –
K
d,l
=0.85
K
d,g
=0.68
Bolle/Seynhaeve Crosby 1D2 10.25 104.85 88
1995 [32] JLT-JOS-15-A
K
d,l
=0.91
K
d,g
=0.96
Modell 10.4 k.A. k.A.
K
d,l
=0.91
K
d,g
=0.96
Lenzing 2001 [34] Leser 25 k.A. k.A.
K
d,l
=0.77
K
d,g
=0.54
Sallet 1984 [33] Kunkle k.A. k.A. k.A.
K
d,l
=0.962
K
d,g
=0.726
Universit¨ at Louvain Leser 28 105 100
1997 [30] K
dl
=0.699
K
dg
=0.521
Bopp & Reuther 20 105 95
K
d,l
=0.780
K
d,g
=0.660
Table 2 Experimental data from
safety valves with initially
sub-cooled two-phase depicted
from literature
Statistical Number Deviation Deﬁnition
variance of absolute deviations X
i,abs
=C
i,exp
−C
i,calc
S
abs
=
¸
n
i=1
X
2
i,abs
n−f −1
variance of relative deviations X
i,rel
=
C
i,exp
−C
i,calc
C
i,exp
S
rel
=
¸
n
i=1
X
2
i,rel
n−f −1
variance of logarithmic deviations X
i,ln
=ln
C
i,exp
C
i,calc
S
ln
=exp
¸
¸
n
i=1
X
2
i,ln
n−f −1
¸
−1
Table 3 Deﬁnition of statistical
numbers used to characterize the
average predictive accuracy of
models (subscript “exp” denotes
experimental values and “calc”
the calculated data)
Fig. 9 Mass ﬂowrateaccordingtotheextendedHNE-DSmethod
and experimentally determined for initially sub-cooled tow-phase ﬂow
through safety valves
–ormoregeneral –non-equilibrium coefﬁcient including
the exponent awill be a good estimate.Nevertheless, fur-
thervalidation withdatausingtheseﬁttingsbeingcarried
out and will be presented elsewhere.
Fig. 10 Flowcoefﬁcient accordingtotheextendedHEMmodel for
initially sub-cooled tow-phase ﬂow through safety valves
7 Conclusion
The HNE-DS model is based on the assumption of homoge-
neous equilibrium ﬂow which is corrected for thermal and
mechanical non-equilibrium effects (see [12] for discussion
1 3
56 Forsch Ingenieurwes (2007) 71: 47–58
of the mechanical equilibrium). It combines the advantages
of both, the well accepted homogeneous equilibrium model
proposed by Leung and the non-equilibriummodel of Henry
andFauske. Besidetheboilingdelayandthephaseslip
insaturatedtwo-phaseﬂowtheHNE-DSmodelhasbeen
extendedtotwo-phasegas/liquidﬂowwithinitiallysub-
cooledliquids. Forthat, theﬂashingdelaycoefﬁcient N,
alreadydeﬁnedtoaccount forboilingdelay,hasbeenex-
tended to take the superheat of an initially sub-cooled liquid
into account. This is deﬁned as a function of the degree of
sub-cooling. Thegeometriceffect onthiscoefﬁcient was
foundtobeof minor importancesaspreviouslyalsore-
ported by Kim et al. [37].
The extended model has been validated with more than
3500 experimental data performed with nozzles and safety
valves with sub-cooled liquid at the inlet. Taking the uncer-
tainty of the measurements into account, the agreement with
the HNE-DS is more than sufﬁcient. Applying the extended
HNE-DS method for sizing nozzles and safety valves will
limit the enormous over-estimation of HEM models which
arecurrentlyrecommendedforexamplebyAPI520and
ISO23521, atlowdegree ofsub-cooling. Thesizeofthe
throttling device will be reduced by a factor of up to 5. Nev-
ertheless, the models given in API 520 and ISO 23521 are
just the boundary values of the HNE-DS model for N =1.
The HNE-DS method included in the draft international
standard ISO/DIS 4126-10 for sizing safety valves for
ﬂashing liquids has been proposed for IEC 60534 (control
valves) and ISO 5167 and ISO 9300 (nozzles, venturis and
oriﬁces). It would be a major advantage for sizing engineers
if the same method and an identical nomenclature were used
for all throttling devices.
A further simpliﬁcation of the proposed HNE-DS method
might be possible, if the search for the maximum ﬂow co-
efﬁcient at critical pressureratiocouldbesubstitutedby
an empirical relation (see Fig. 3). Additional investigations
arerecommendedtocombinetheproposedexponentsof
the non-equilibriumcoefﬁcient for certain throttling devices
into a single correlation for all throttling devices.
Thenon-equilibriumeffectrecommendedintheHNE-
DSmethodwasdeterminedbymeansofaregressionan-
alysis of experimental data from various authors in the lit-
erature. A more academic solution, where the exponent a is
based on physical principles like depressurization rate and
bubble growth models would further improve the method.
Appendix
Example calculation (MathCad Version 12)
Sizing of a safety valve
Example:Ventingofa10 m
3
reactor(TEMPEREDSYS-
TEM)
Input data:
p
0
:=10 bar sizing pressure (inlet pressure)
p
b
:=1 bar back pressure
Q
m
:=25000 kg/h mass ﬂow rate to be discharged
according to ISO 4126-10
x
0
:=0 inlet mass ﬂow quality
Temperaturedeterminedbyreactioncalorimetryand
property data of the reactor inventory at inlet condition
T
0
:=453.05 K temperature in the pressurized
system at sizing conditions
p
sat
(T
0
) :=9.5 bar pressure at saturation condition
cp
l0
:=4650 J/kg K speciﬁc heat capacity (liquid phase)
∆h
v0
:=1826000J/kglatent heat of vaporization
v
l0
:=0.001193 m
3
/kg speciﬁc volume liquid phase
v
g0
:=0.1984 m
3
/kg speciﬁc volume gas phase
v
0
:= x
0
· v
g0
+(1−x
0
) · v
l0
speciﬁc volume of reactor inventory
v
0
=1.193×10
−3
m
3
/kg
Certiﬁedderateddischarge coefﬁcients of the safety
valve (given by valve manufacturer)
K
dg
:=0.77 certiﬁed derated discharge coefﬁcient for
single-phase gas/vapor ﬂow
K
dl
:=0.5 certiﬁed derated valve discharge coefﬁcient
for single-phase liquid ﬂow
Calculation of the dischargeable mass ﬂux
through a safety valve
(two-phase gas-liquid mixture)
η
s
:=
p
sat
(T
0
)
p
0
η
s
=0.95 ratio of the saturation pressure
to sizing pressure
η
b
:=
p
b
p
0
η
b
=0.1 ratio of back pressure to the
sizing pressure
Maximum search for maximum ﬂow coefﬁcient
and critical pressure ratio
(Deﬁnition of vector parameters)
Steps :=100 Number of calculation steps
Interval :=
1−η
b
Steps−1
Step size of pressure ratio
j :=0, 1 . . . (Steps −1) Index parameter running from
0 number of steps deﬁned
η
j
:=1−Interval· j Pressure ratio at each step
N
j
:=
x
0
+cp
l0
· p
0
· η
s
· T
0
·
v
g0
−v
l0
∆h
2
v0
ln
η
s
η
j
η
−0.6
s
Non-equilibriumcoefﬁcient at
each step
ω
j
:=
1
x
·
x
0
·v
g0
v
0
+
cp
l0
·T
0
· p
0
·η
s
v
0
v
g0
−v
l0
∆h
v0
2
· N
j
Compressibility coefﬁcient at
each step
1 3
Forsch Ingenieurwes (2007) 71: 47–58 57
Fig. 11 Flow coefﬁcient versus
pressure ratio
C
j
:=
(1−η
s
)+
¸
ω
j
·η
s
·ln
ηs
η
j
−(ω
j
−1)(η
s
−η
j
)
¸
ω
j
ηs
η
j
−1
+1
Flow coefﬁcient at each step
Ccrit :=max(C) Ccrit =0.465
Maximum of ﬂow coefﬁcient
max :=j ←0
while
C
j
≤Ccrit · 0.9999
j ←j +1
j
Step where maximum ﬂow
coefﬁcient occurs
η
crit
:=η
max
η
crit
=0.691 critical pressure ratio
Flow coefﬁcient versus pressure ratio (see Fig. 11)
Critical pressure ratio exceeds back pressure ratio
Data at critical pressure ratio are taken to calculate the ﬂow
coefﬁcient
η :=η
max
η =0.691 pressure ratio
N := N
max
N =0.034 boiling delay factor
ω :=ω
max
ω =0.666 compressibility coefﬁcient
C :=C
max
C =0.465 ﬂow coefﬁcient
Estimationof thetwo-phasedischargecoefﬁcient of the
safety valve
:=1−
v
l0
v
0
¸
ω
ηs
η
+1
¸
=0.2
void fraction in the throat area of the valve
K
d2ph
:= K
dg
·+(1−) · K
dl
K
d2ph
=0.554
derated two-phase discharge coefﬁcient of the safety valve
m
SV
:= K
d2ph
· C
2p
0
v
0
m
SV
=1.055×10
4
kg/m
2
s
dischargeable mass ﬂux through the safety valve
A
SV
=
Q
m
m
SV
A
SV
=6.581×10
−4
m
2
minimumrequired cross sectional area of the safety valve
d
SV
:=
4
π
A
SV
d
SV
=28.9 mm
minimum required diameter of the safety valve
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1 3