DOI 10.1007/s10010-006-0043-3

O R I G I N A L A R B E I T E N · O R I G I N A L S

Forsch Ingenieurwes (2007) 71: 47–58

Sizing of nozzles, venturis, orifices, control and safety valvesfor initially sub-cooled gas/liquid two-phase flow –The HNE-DS method

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 flowmodels, flow coefficients and nomenclature. They are gener-ally valid only for single-phase gas and liquid flow. Commonto all is the concept of one-dimensional nozzle flow in combi-nation with a correction factor (e.g. the discharge coefficient)to correct for non-idealities of the three-dimensional flow.With the proposed partial non-equilibrium HNE-DS methodan attempt is made to standardize all sizing procedures byan 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-cooledliquids entering the throttling device. To account for non-equilibrium effects, i.e. superheated liquid due to rapid de-pressurisation, the non-equilibrium coefficient used in theHNE-DS method is adapted to those inlet flow conditions.A comparison with experimental data demonstrates the goodaccuracy of the model.

Auslegung von Dusen, Venturis, Blenden, Stell-und Sicherheitsventilen fur eingangs unterkuhlteGas-Flussigkeits-Stromungen nach der HNE-DSMethode

Zusammenfassung Die derzeitigen Regelwerke zur Ausle-gung von Dusen, Venturies, Blenden, Stell- und Sicherheits-ventilen basieren auf verschiedenen Stromungsmodellen,Durchflusskoeffizienten und sind mit verschiedenen Nomen-klaturen beschrieben. Sie gelten nur fur Einphasenstromung

J. Schmidt (�)BASF AG, GCT/S-L511,Ludwigshafen, Germanye-mail: juergen.schmidt@onlinehome.de

von Gasen und Flussigkeiten. Gemeinsam ist den Modellenin den Regelwerken die Kombination aus einer idealisier-ten Dusenstromung und einem Korrekturfaktor (z.B. demAusflusskoeffizienten), um die Nicht-Idealitaten der dreidi-mensionalen realen Stromung zu korrigieren. Die neue HNE-DS Methode, ein Dusen-Stromungsmodell mit Gasen undFlussigkeiten im partiellen Ungleichgewicht, erlaubt es, diebestehenden Auslegungsempfehlungen in den verschiedenenRegelwerken zu vereinheitlichen und gleichzeitig auf Zwei-phasenstromungen zu erweitern. Die HNE-DS Methode, diezunachst fur siedende und nicht-verdampfende Zweiphasen-stromungen entwickelt worden ist, wird erweitert fur an-fangs unterkuhlte Flussigkeiten im Eintritt der Armaturen.Ungleichgewichtseffekte, beispielsweise die Uberhitzungder Flussigkeit bei schnellem Druckabfall, werden mit ei-nem erweiterten Ungleichgewichtsfaktor im HNE-DS Mo-dell berucksichtigt. Der Vergleich mit experimentellen Datenzeigt die gute Genauigkeit des Modells.

List of symbols

Variable Unit Definitiona – exponent of the non-equilibrium

coefficient NA m2 cross-sectional area of the nozzle throat

(seat aera of valve)C – flow coefficientCcrit – flow coefficient at critical pressure

ratio in the nozzle throatcpi,0 J/(kg K) specific liquid heat capacity at inlet

conditionsd m nozzle throat diameterd0 m nozzle inlet diameterKd,2ph – derated two-phase flow valve

discharge coefficient

1 3

48 Forsch Ingenieurwes (2007) 71: 47–58

Kd,g – certified (derated) valve dischargecoefficient for single-phasegas/vapor flow

Kd,l – certified (derated) valve dischargecoefficient for single-phase liquidflow

lPipe m length of piping behind the nozzlethroat with an diameter equal to thenozzle throat diameter

m kg/(m2s) mass fluxN – non-equilibrium coefficientp Pa pressure in the nozzle throatp0 Pa nozzle inlet pressureps(T0) Pa saturation pressure at inlet

temperaturepb Pa back pressurepc Pa thermodynamic critical pressureQm kg/s mass flow rate through the nozzleT0 K nozzle inlet temperatureTc K thermodynamic critical temperaturev m3/kg specific volume in the nozzle throatv0 m3/kg specific volume in the nozzle inletv∗ m3/kg dimensionless specific volumex0 – mass flow quality in the nozzle inletxeq – mass flow quality in the nozzle throat

under thermodynamic equilibriumconditions

∆xeq – change of mass flow quality betweennozzle inlet and throat underthermodynamic equilibriumconditions

ε – void fraction in the nozzle throatβ – diameter ratioη – pressure ratioηb – ratio of back pressure to the inlet

pressureηcrit – critical pressure ratioηS – ratio of the saturation pressure

corresponding to the nozzle inlettemperature (measure of liquidsubcooling) to the inlet pressure

κ – Isentropic coefficientλinsul W/(m2K) heat transfer coefficient of the

insulationω – compressibility coefficientω(N) – compressibility coefficient depending

on the non-equilibrium coefficient Nωeq – compressibility coefficient for

a homogeneous fluid underthermodynamic conditions,ω (N = 1)

∆hv,0 J/kg latent heat of vaporization at inletcondition

1 Introduction

Two-phase mass flow rates through throttling devices aregenerally calculated based on simplified geometries. Mostoften a nozzle is considered. The result is then corrected byan experimentally determined factor, i.e. a friction or dis-charge coefficient, to account for any deviation in the flowdue to the real geometry – an orifice, a venturi or valve. Cur-rently the only standards which exist are for the sizing ofsafety valves given in API 520 [1] and ISO 23521 [2]. Theseare based on the world wide accepted ω-method developedby J.C. Leung, [3, 4], and recommended by the DIERS In-stitute [5]. Several classical sizing text books also makereference to it [6, 7]. One of the major advantages of thismethod is its use of known or easily measurable propertydata at inlet stagnation condition of the safety valve.

The use of the ω-method for saturated two-phase flowgenerally leads to conservative sizing results, because ho-mogeneous equilibrium flow through the safety valve isassumed. However, if an initially sub-cooled or a boilingliquid with a low mass flow quality has to be consideredat the inlet of a safety valve, it is well known, that themethod provides an in-acceptable over-estimation of the re-quired size [e.g. 8,9]. As an alternative, the Henry/Fauskemodel [10] which is based on a more accurate fit to ex-perimental data can be applied in such situations to cal-culate the mass flow rate. Henry and Fauske also pro-posed a boiling delay factor to account for the thermalnon-equilibrium of the fluid and get excellent results whencompared with the flow through nozzles. Unfortunately, themodel of Henry/Fauske is based on physical property datawhich are only rarely available in industry and additionallyit is more complicate to use than the ω-method.

To overcome the conservatism of the ω-method for lowquality inlet flow, both the ω-method and the Henry/Fauskemodel are combined into the HNE-DS method (Homoge-neous Non-Equilibrium Method of the authors Diener andSchmidt) to account for thermodynamic non-equilibrium ef-fects [11, 12]. Mechanical non-equilibrium effects are ac-counted for by means of a slip model. In this way, theprevious work of J.C. Leung and Henry and Fauske is rec-ognized and engineers in practice may continue to use theirtraditional methods, like API 520.

In addition to the sizing of safety valves, a generaliza-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 2006as a draft international standard, a preliminary standard hasbeen published in [13, 14]) and is proposed for inclusionin IEC 60534 (control valves) [16]. Additionally, it is rec-ommended to extend ISO 5167 [17] and ISO 9300 [18](nozzles, orifices, venturis) for two-phase flow.

1 3

Forsch Ingenieurwes (2007) 71: 47–58 49

In the following, the HNE-DS method is derived andextended 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-tling device as a frictionless, adiabatic nozzle. The fluidis assumed to be a quasi single-phase, i.e. a homogeneousmixture of gas and liquid in equilibrium, with two-phaseproperties. Correction of the simplified model are definedfor non-idealities like boiling delay and slip between gasand liquid phase, which may be characteristic for certainthrottling devices. Those non-idealities are induced, e.g., bya contraction and redirection of the flow and due to fric-tion and wall heat exchange. The more precise the nozzleflow model accounts for non-equilibrium effects and realproperties of the fluids, the fewer dependencies have to betaken into consideration for a discharge coefficient. In anycase, the discharge coefficient must be 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 encounteredin industry.

The one-dimensional momentum balance for the flowthrough a frictionless, adiabatic nozzle with no gravity ef-fects encountered is,

C =

√√√√√√

−η∫

η0

v∗dη

(v∗)2 −β4; η = p

p0; v∗ = v

v0; β = d

d0. (1)

The nomenclature of ISO/DIS 4126-10 is identically appliedin the present paper.

Herein, η is the ratio of the pressure in the nozzle throat pand the inlet p0 (symbols without subscripts refer to the noz-zle throat while the subscript ,,0“ stands for the inlet of thenozzle), v∗ is the specific volume ratio and β the diameterratio. The flow coefficient C is, by definition, the normalizedmass flow rate Qm through the nozzle,

C = Qm

A ·√

2 · p0v0

; A = π

4d2 . (2)

Equation 1 is valid for single phase gas and liquid flow aswell as for two-phase mixtures. The gas/liquid two-phaseflow is treated as quasi single-phase flow with a specific vol-ume of a mixture. Any information about interfacial heatand mass transfer between the phases is included in the di-mensionless specific volume of the flow v∗. Its time and

cross-sectional average for a homogeneous two-phase flowunder thermodynamic equilibrium condition is [5, pp. 58ff],

v∗ = x · vg

v0+ (1 − x) · vl

v0, (3)

from where an equation of state v∗(η) for the profile alongthe nozzle may be determined by integration of the deriva-tive,

v0dv∗

dη= (

vg −vl) · dx

dη+ x · dvg

dη+ (1 − x) · dvl

dη(4)

The liquid may be considered as almost incompressible,the gas as following the ideal gas law and the boiling lineof the gas/liquid mixture may be prescribed by the Clau-sius Clapeyrons law. Heat of vaporization and liquid spe-cific volume are constant and the temperature of the gasis taken equal to that of the liquid (spontaneous heat ex-change). What remains is the information about the changeof mass flow quality along the pressure curve in the nozzle,i.e. the interfacial heat and mass transfer. Any heat transferlimitation due to steep pressure gradients within the noz-zle will result into a thermodynamic non-equilibrium – orboiling delay – of the flow. Henry and Fauske [10] iden-tified the boiling delay as a deviation of the mass flowquality from equilibrium conditions at a certain pressuredrop

dx

dη= dxeq

dη· N . (5)

The proposed dependency of the boiling delay factor N onthe mass flow quality is linear at low qualities and constantfor mass flow qualities larger than 0.14. In contrast, Dienerand Schmidt [11, 12] suggested a (continuous) power-lawfunction for the boiling delay factor. The basis of their func-tion is the mass flow quality in the narrowest cross sectionof the nozzle xeq – the nozzle throat – if both the vapor andliquid phase are in thermal equilibrium:

N = [

xeq]a ; a ∈ 0 . . .∞ ; xeq ∈ 0 . . .1 ; ⇒ N ∈ 0 . . . 1 .

(6)

The exponent “a” depends on the relaxation time for thetwo-phase flow in the nozzle up to its narrowest cross sec-tion. In a very short nozzle, the momentum and heat ex-change between both phases is poor and, hence, the boilingdelay reaches its maximum (a →∝; N → 0). The flow is al-most frozen. If vapor and liquid would have time to reach itsequilibrium state, i.e. in a very long nozzle, almost no boil-ing delay will occur (a → 0; N → 1). As a general rule, thelarger the inlet mass flow quality is, the less pronounced isthe boiling delay effect, Fig. 1.

1 3

50 Forsch Ingenieurwes (2007) 71: 47–58

Fig. 1 Mass flux calculated with the HEM and the Frozen Flowmodel versus inlet mass flow quality for a nozzle investigated bySozzi/Sutherland [22]

Following the derivation of the original ω-method [3]and taking the boiling delay coefficient N into considerationyields an equation of state for a two-phase flow includingthermal non-equilibrium effects,

v∗ = ω(N)

(1

η− 1

η0

)

−1 , (7)

where the compressibility coefficient is defined by,

ω(N) = 1

κ

x0 ·vg0

v0+ cpl0 · T0 · p0 ·η0

v0·[vg0 −vl0

∆hv0

]2

· N , (8)

N =(

x0 + cpl0 · T0 · p0 ·η0 ·(

vg0 −vl0

∆h2v0

)

· ln(

η0

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-mal change of state in single phase gas flow compared toa two-phase gas liquid flow. The compressibility coefficientω(N) leads to the original ω-parameter, when vapor and li-quid phase are in thermal equilibrium (N = 1; ω(N = 1) =ωeq). If there is no mass and heat transfer between vaporand liquid at all (frozen flow) or in a non-flashing gas/liquidflow, the second term on the right hand side of Eq. 8 wouldvanish (N = 0).

Due to simplified assumptions, such as constant heat ofvaporization, Eq. 7 results in large unacceptable uncertain-ties close to the thermodynamic critical point of a fluid.Hence, it should only be used, if the reduced inlet pres-sure or temperature of the fluid do not exceed values ofp0/pc ≤ 0.5 or T0/Tc ≤ 0.9, respectively. Additionally, formulti-component fluids the largest boiling temperature dif-ference of two compounds should be less than 100 ◦C. Inany other case, a more precise equation of state should beconsidered [7, 19].

3 Initially sub-cooled liquid flow

If a sub-cooled liquid enters a nozzle three distinct void pro-files may develop up to the nozzle throat, Fig. 2: (I) pureliquid flow throughout the nozzle or (II) just flashing inthe nozzle throat (highly sub-cooled two-phase flow) and(III) flashing prior to nozzle throat (low sub-cooled two-phase flow). Let pb be the back pressure downstream of thenozzle throat, then the flow coefficient for pure liquid flowyields (profile I)

Cl =√

1 −ηb

1 −β4; ηb = pb

p0. (10)

Flashing of the liquid in the nozzle throat will cause a localchoke. A first estimate for the throat pressure is the satura-tion pressure of the liquid at inlet temperature pS(T0), theflow coefficient becomes (profile II)

C =√

1 −ηS

1 −β4; ηS = pS(T0)

p0. (11)

Only if gas and liquid phase are homogeneously mixedand in thermodynamic equilibrium can the flow coefficient,Eq. 11, be reached. Superheating of the liquid phase typ-ically causes larger mass flow rates. This effect can beaccounted for by a flashing delay – or more general –non-equilibrium coefficient N as already proposed for theHNE-DS model in saturated two-phase flow. According toLeung [20] the integral in Eq. 1 should be separated intoa flow region for liquid flow up to ηS (s. Eq. 11) and intoa two-phase region starting at η0 = ηS until the nozzle throatto yield a generally valid flow coefficient for single-phasegas and liquid flow as well as for homogeneous two-phase

Fig. 2 Void profiles ininitially sub-cooledtwo-phase flow througha nozzle

1 3

Forsch Ingenieurwes (2007) 71: 47–58 51

gas/liquid mixtures,

C =

√√√√√√

(1 −ηS)−η∫

ηS

v∗ (ω(N)) ·dη

(v∗ (ω(N)))2 −β4; ηS = pS(T0)

p0. (12)

The pressure ratio η, i.e. the pressure in the nozzle throat tothe inlet pressure, which is used in Eq. 7 to define v∗ and inEq. 9 to calculate N, equal the back pressure ratio η = ηb

in case of sub-critical flow and the critical pressure ratioη = ηcrit, if the flow in the nozzle throat is choked. The crit-ical pressure ratio ηcrit is defined as the ratio where the flowcoefficient C(η), (s. Eqs. 12, 7–9), reaches its maximumvalue.

For a plenum inlet flow (β = 0) and a non-equilibriumcoefficient N independent of the pressure ratio η, Eq. 12would lead to an analytical solution for the flow coefficient

C =√

(1 −ηS)+[

ω(N) ·ηS · ln(

ηSη

)

− (ω(N)−1) (ηS −η)]

ω(N)(

ηSη

−1)

+1.

(13)

Overall, Eqs. 12, 7–9 (integral solution) or Eqs. 13, 8, 9(analytical solution) are applicable for all flow conditionstypically encountered in industry.

Flow characteristic

Single phase liquid x0 = 0; v∗ = 1; η0 = 1; η = ηbSingle phase gas x0 = 1; v∗ = vg/vg,0;

η0 = 1; η ≥ ηcrit & η ≥ ηbInitially sub-cooledtwo-phase flow

x0 = 0; η0 = ηS; η ≥ ηcrit & η ≥ ηb

Saturatedtwo-phase flow

x0 ≥ 0; η0 = 1; η ≥ ηcrit & η ≥ ηb

Non-flashingtwo-phase flow

x = x0 = const;η0 = 1; η ≥ ηcrit & η ≥ ηb; N ≡ 1

4 Critical mass flow rate

The critical mass flow rate is defined as the maximum flowrate through the nozzle for given inlet conditions,

dC

dη= 0 ⇒ max[C(η); η ∈ ηb, 1] , (14)

and is determined most accurately by integration of Eq. 12or using Eq. 13 and a subsequent maximum search, Eq. 14.At its maximum, either the back pressure or the critical pres-sure is reached in the nozzle throat. Typical solutions ofEq. 14 for C are presented in Fig. 3 for a constant value ofthe exponent a. A sub-cooling of ηS = 1 represents an ini-tially saturated liquid and a value of ηS = 0.5 gives typicalresults for initially high sub-cooled liquids, see Eq. 10.

Fig. 3 Flow coefficient according to the HNE-DS model as a functionof pressure ratio for certain degree of initially sub-cooled liquids

An analytical solution of Eq. 14 may be performed fora plenum flow (β = 0) of a homogenous mixed fluid in ther-modynamic equilibrium as proposed by Leung [20]

Ccrit,HEM =√

(1 −ηS)+[

ωeq ·ηS · ln(

ηSη

)

− (ωeq −1)(ηS −η)]

ωeq

(ηSη

−1)

+1(15)

Herein, ωeq is the compressibility coefficient for N = 1(see Eqs. 8 and 9, ω(N = 1) = ωeq). In Eq. 15 is η = ηcrit

if ηcrit ≤ ηS (low sub-cooling), otherwise η = ηS (high sub-cooling) leading to

Ccrit,HEM = √

1 −ηS . (16)

The critical pressure ratio ηcrit at the critical flow coefficientwas derived by applying Eq. 14 yielding a transcendentalequation

0 = ω2eq −2ωeq +1

2ωeqηSη2

crit −2(ωeq −1)ηcrit

+ωeq ln

(ηcrit

ηS

)

+ 3

2ωeqηS −1 . (17)

Equations 15 and 17 are good approximations for two-phaseflow with saturated mixtures (ηs = 1). In case of sub-cooledliquids the results are poor, because the critical pressureratio is highly overestimated. There is a strong dependenceof the flow coefficient on the nozzle length and the degree ofsub-cooling as can be seen from the well known experimen-tal data of Sozzi and Sutherland [22], Fig. 4. Additionally,the low sub-cooled region is limited to very high satura-tion pressure ratios ηS, typically in the range of 0.9 to 1, i.e.to very small sub-cooling temperatures, Fig. 5. At low in-let pressures even 1 K makes the difference between the lowsub-cooled region (flashing within the nozzle) and the highsub-cooled region (flashing in the nozzle throat).

1 3

52 Forsch Ingenieurwes (2007) 71: 47–58

Fig. 4 Uncertainty of HEM model compared to experiments fromSozzi and Sutherland [22] for initially sub-cooled steam/water flowthrough nozzles of different length

Fig. 5 Sub-cooling of liquid at nozzle entrance versus nozzle inletpressure for experiments given in the open literature

Most of the literature data are measured in the high sub-cooling region and the flow rates calculated based on theHEM assumption are significant too low. As a consequenceof the small region of low sub-cooling, the thermodynamicequilibrium model of Eq. 15 instead of Eq. 14 is most of-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 coefficientN

In the literature there are at least two general methods toaccount for the thermal non-equilibrium in flashing flows:one method is based on the growth of a single bubble fol-lowing a certain pressure drop [23]. Due to the lack of anydata, the total number of nuclei in a liquid has to be definedempirically and the models are highly sensitive to this pa-rameter. Therefore, these types of models can not be appliedfor industrial purposes. The second type of model is based

on rapid depressurization experiments due to pipe rupturesinvestigated for the nuclear industry [24, 25]. Within mil-liseconds the pressure falls locally very much below the sat-uration pressure, and is followed by a vapor explosion. Thedepressurization gradient within those experiments is muchlarger than in typical throttling devices. Hence, this theoryis also not applicable for calculating the non-equilibriumeffects. Overall, there is no physical method available to ac-count for the thermal non-equilibrium effects which occur innozzles, venturis, orifices or valves.

In the HNE-DS method an alternative approach is pro-posed, based on a semi-empirical non-equilibrium coeffi-cient N. Physically, the coefficient N is a measure of therelaxation time to exchange heat and mass between bothphases and depends on the geometry of the throttling de-vice as well as on the distribution of gas and liquid phase.It therefore represents both the degree of super-saturation inthe sub-cooled liquid region and boiling delay in saturatedtwo-phase flow,

N = [

x0 +∆xeq,nozzle]a ; N ∈ 0 . . . 1 . (18)

The exponent “a” is derived from experimental nozzle flowdata. The correlation with experimental measurements isan ongoing process, depending on the data available in theliterature.

Diener and Schmidt [11, 12] proposed a value of a = 2/5for safety valves and a = 3/5 for control valves for saturatedtwo-phase gas/liquid flow by comparison with limited ex-perimental data from valves. Good agreement for inlet massflow qualities larger than 0.05 has been shown. Due to thelack of more detailed data, no recommendations have beengiven for lower mass flow qualities or initially sub-cooledtwo-phase flow.

In an initially sub-cooled two-phase flow the flashing de-lay or non-equilibrium coefficient N depends on the degreeof sub-cooling. This is shown by the comparison in Fig. 6of the flow coefficient for homogeneous equilibrium flowCcrit,HEM, Eq. 14, and experimentally determined flow co-efficients Cexp, Eq. 2. The lower the degree of sub-coolingis (ηS → 1), the larger is the deviation from HEM. Ex-perimentally determined values of the flow coefficient areup to 5 times larger, than calculated values based on theHEM model. This is equally true for nozzles and for safetyvalves.

The exponent a in Eq. 18 for sub-cooled two-phase flow(x0 ≡ 0) was determined by a regression analysis of lit-erature data measured using nozzles and safety valves asfollows

a = 7.5lpiped0

+7.5· (ηS)

−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, lpipe is the length of pipe with a diameter equal to thenozzle throat diameter behind the nozzle throat1. For noz-zles without a pipe tail, Eq. 20, reduces to

a = (ηS)−0.6 . (20)

In case of a saturated gas/liquid two-phase flow (x0 ≥ 0;η0 = 1; η ≥ ηcrit) Eqs. 13, 8, 9 reduces to the HNE-DS modelalready proposed by Diener and Schmidt [11, 12].

6 Validation of the extended HNE-DS model

Literature data from nozzles, Table 1 [22, 26–29], have beenused for the comparison of the HNE-DS model extended toinitially sub-cooled liquids with experimental results. Thereis no unambiguous tendency concerning the nozzle diameterand length. Additionally, even very small deviations of massflow quality result in large deviations of the mass flow rate atlow sub-cooling. Physically, any gas dissolved in the liquidphase or absorbed on the wall surface of the nozzle may actas a nucleation source. Hence, the experimental data oftenhas a high degree of uncertainty.

The HNE-DS model extended to initially sub-cooledliquids by means of the non-equilibrium coefficient Nleads to fairly good results in comparison to experimen-tal data. Figure 7 shows the overall tendency dependingon the degree of sub-cooling compared to the data ofLee [27]. Especially at low sub-cooling the model is byfar better than the HEM model. A frozen flow assump-tion (no vaporization) would give highly overestimatedresults.

Figure 8 shows the comparison of the HNE-DS modelwith the ∼ 1500 investigated nozzle data with sub-cooledliquids at the inlet. Considering the uncertainty of the ex-

1 Nozzles with pipes of up to 500 mm length have been considered inthis study

Fig. 7 Comparison of flow coefficients measured by Lee [27] andcalculated with the HNE-DS model extended for initially sub-cooledliquids

Fig. 8 Mass flow rate according to the extended HNE-DS methodand experimentally determined for initially sub-cooled tow-phase flowthrough nozzles

perimental data the HNE-DS model is in good agreementwith the measurements. Of course, it is still possible to fur-ther improve the correlation, but that would lead to a morecomplex function of the exponent a in the model, beside thefact, that there are not detailed data available about the influ-ence of the different parameters.

Beside the nozzle data, roughly 2000 measurementsfrom safety valves, Table 2 [30–33], have also been com-pared with the HNE-DS model. As proposed by Diener andSchmidt [11], the valve discharge coefficient for two-phaseflow Kd,2ph was based on the discharge coefficients for gasflow Kd,g and liquid flow Kd,l , in general given by the valvemanufacturer,

Kd,2ph = ε · Kd,g + (1 − ε) · Kd,l . (21)

where ε is the void fraction in the narrowest flow crosssection. Diener and Schmidt [11, 12] proposed to use the

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 fluidSource Type [mm] [mm] [mm] [bar] [◦C]

Sozzi, Sutherland No 1 12.7 44.5 114 54–69 220–285 demineralized1975 [22] Venuri outlet conus water

No 2 12.7 44.5 0Rounded Nozzle 12.7with Tail Pipe 38.1

63.5190.5317.5508635

1778No 4 12.7 – 4.7Sharped edged nozzle 195.2

322520.7639

No 5 19 44.5 –Rounded NozzleNo 6 54 732 380Venturi outlet conusNo 6 76.2 696 380Venturi outlet conusNo 7 28 63.5 165Venturi outlet conus

Boivin 1979 [26] Nozzle with tail pipe 12 50 450 20–90 200–300 water30 130 160050 130 1700

Veneau 1992 [29] Nozzle with tail pipe 2 2 1.2 280–300 60–120 propane5 6 3

Lee, Swinnerton Nozzle Sharped 1.8 – 1.8 140–300 200–430 water1983 [27] Edged Inlet

Nozzle rounded Inlet 1.8 1.3 5.3Nozzle rounded Inlet 2.5 1.3 7.6(outlet guide)Nozzle rounded Inlet 2.5 1.3 7.6Nozzle rounded Inlet 2.8 1.3 8.4

Simoneau, Hendricks Venturi outlet conus 3.555 74 237 6–140 −244–12 nitrogen1984 [28] methane

2 D nozzle Hohe: 1.09 205 106Breite: 10.1

Venturi outlet conus 2.934 7.8 54Venturi outlet conus 3.555 237 74Duse 2 6.4 k.A. k.A.Duse 3 4 k.A. k.A.

critical pressure ratio η = ηcrit in Eq. 22 under thermody-namic equilibrium conditions for simplicity.

ε = 1 − vl,0

v0 ·[

ω ·(

1η−1

)

+1] , (22)

The HNE-DS model gives excellent agreement with themeasured valve data even for initially sub-cooled two-phaseflow as shown in Fig. 9. The mean logarithmic deviation– see definition Table 3 – is only 16%. The HEM modelrecommended by API 520 and ISO 23521 is less accu-rate. The experiments are highly under-estimated, Fig. 10,

with a mean logarithmic deviation of 78%. Even the modelof Darby [35] tends, in general, to underestimate the ex-perimental data. The reason could be that the model hasonly been fitted to a certain nozzle used by Sozzi andSutherland.

An overall comparison of the HNE-DS model with morethan 4000 data including the data with saturated two-phaseflow and non-flashing flow is given in [36].

The HNE-DS model can equally applied to controlvalves, orifices and other throttling devices. For initiallysub-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 lengthSource [mm] [mm] [mm]

Boccardi 2005 [31] Leser 10 – –Kd,l = 0.85Kd,g = 0.68

Bolle/Seynhaeve Crosby 1D2 10.25 104.85 881995 [32] JLT-JOS-15-A

Kd,l = 0.91Kd,g = 0.96Modell 10.4 k.A. k.A.Kd,l = 0.91Kd,g = 0.96

Lenzing 2001 [34] Leser 25 k.A. k.A.Kd,l = 0.77Kd,g = 0.54

Sallet 1984 [33] Kunkle k.A. k.A. k.A.Kd,l = 0.962Kd,g = 0.726

Universitat Louvain Leser 28 105 1001997 [30] Kdl = 0.699

Kdg = 0.521Bopp & Reuther 20 105 95Kd,l = 0.780Kd,g = 0.660

Table 2 Experimental data fromsafety valves with initiallysub-cooled two-phase depictedfrom literature

Statistical Number Deviation Definition

variance of absolute deviations Xi,abs = Ci,exp −Ci,calc Sabs =√

∑ni=1 X2

i,absn− f −1

variance of relative deviations Xi,rel = Ci,exp−Ci,calcCi,exp

Srel =√

∑ni=1 X2

i,reln− f −1

variance of logarithmic deviations Xi,ln = lnCi,expCi,calc

Sln = exp

{√∑n

i=1 X2i,ln

n− f −1

}

−1

Table 3 Definition of statisticalnumbers used to characterize theaverage predictive accuracy ofmodels (subscript “exp” denotesexperimental values and “calc”the calculated data)

Fig. 9 Mass flow rate according to the extended HNE-DS methodand experimentally determined for initially sub-cooled tow-phase flowthrough safety valves

– or more general – non-equilibrium coefficient includingthe exponent a will be a good estimate. Nevertheless, fur-ther validation with data using these fittings being carriedout and will be presented elsewhere.

Fig. 10 Flow coefficient according to the extended HEM model forinitially 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 andmechanical non-equilibrium effects (see [12] for discussion

1 3

56 Forsch Ingenieurwes (2007) 71: 47–58

of the mechanical equilibrium). It combines the advantagesof both, the well accepted homogeneous equilibrium modelproposed by Leung and the non-equilibrium model of Henryand Fauske. Beside the boiling delay and the phase slipin saturated two-phase flow the HNE-DS model has beenextended to two-phase gas/liquid flow with initially sub-cooled liquids. For that, the flashing delay coefficient N,already defined to account for boiling delay, has been ex-tended to take the superheat of an initially sub-cooled liquidinto account. This is defined as a function of the degree ofsub-cooling. The geometric effect on this coefficient wasfound to be of minor importances as previously also re-ported by Kim et al. [37].

The extended model has been validated with more than3500 experimental data performed with nozzles and safetyvalves with sub-cooled liquid at the inlet. Taking the uncer-tainty of the measurements into account, the agreement withthe HNE-DS is more than sufficient. Applying the extendedHNE-DS method for sizing nozzles and safety valves willlimit the enormous over-estimation of HEM models whichare currently recommended for example by API 520 andISO 23521, at low degree of sub-cooling. The size of thethrottling device will be reduced by a factor of up to 5. Nev-ertheless, the models given in API 520 and ISO 23521 arejust the boundary values of the HNE-DS model for N = 1.

The HNE-DS method included in the draft internationalstandard ISO/DIS 4126-10 for sizing safety valves forflashing liquids has been proposed for IEC 60534 (controlvalves) and ISO 5167 and ISO 9300 (nozzles, venturis andorifices). It would be a major advantage for sizing engineersif the same method and an identical nomenclature were usedfor all throttling devices.

A further simplification of the proposed HNE-DS methodmight be possible, if the search for the maximum flow co-efficient at critical pressure ratio could be substituted byan empirical relation (see Fig. 3). Additional investigationsare recommended to combine the proposed exponents ofthe non-equilibrium coefficient for certain throttling devicesinto a single correlation for all throttling devices.

The non-equilibrium effect recommended in the HNE-DS method was determined by means of a regression an-alysis of experimental data from various authors in the lit-erature. A more academic solution, where the exponent a isbased on physical principles like depressurization rate andbubble growth models would further improve the method.

Appendix

Example calculation (MathCad Version 12)

Sizing of a safety valve

Example: Venting of a 10 m3 reactor (TEMPERED SYS-TEM)

Input data:p0 := 10 bar sizing pressure (inlet pressure)pb := 1 bar back pressureQm := 25 000 kg/h mass flow rate to be discharged

according to ISO 4126-10x0 := 0 inlet mass flow quality

Temperature determined by reaction calorimetry andproperty data of the reactor inventory at inlet condition

T0 := 453.05 K temperature in the pressurizedsystem at sizing conditions

psat(T0) := 9.5 bar pressure at saturation conditioncpl0 := 4650 J/kg K specific heat capacity (liquid phase)∆hv0 := 1 826 000 J/kg latent heat of vaporizationvl0 := 0.001193 m3/kg specific volume liquid phasevg0 := 0.1984 m3/kg specific volume gas phasev0 := x0 ·vg0 + (1 − x0) ·vl0

specific volume of reactor inventoryv0 = 1.193 ×10−3 m3/kg

Certified derated discharge coefficients of the safetyvalve (given by valve manufacturer)

Kdg := 0.77 certified derated discharge coefficient forsingle-phase gas/vapor flow

Kdl := 0.5 certified derated valve discharge coefficientfor single-phase liquid flow

Calculation of the dischargeable mass fluxthrough a safety valve(two-phase gas-liquid mixture)

ηs := psat(T0)

p0ηs = 0.95 ratio of the saturation pressure

to sizing pressureηb := pb

p0ηb = 0.1 ratio of back pressure to the

sizing pressure

Maximum search for maximum flow coefficientand critical pressure ratio(Definition of vector parameters)

Steps := 100 Number of calculation stepsInterval := 1−ηb

Steps−1 Step size of pressure ratioj := 0, 1 . . . (Steps−1) Index parameter running from

0 number of steps definedηj := 1 − Interval· j Pressure ratio at each step

Nj :=(

x0 + cpl0 · p0 ·ηs · T0 · vg0−vl0

∆h2v0

ln(

ηsηj

))η−0.6

s

Non-equilibrium coefficient ateach step

ωj := 1x · x0 ·vg0

v0+ cpl0 ·T0·p0 ·ηs

v0

(vg0−vl0∆hv0

)2 · Nj

Compressibility coefficient ateach step

1 3

Forsch Ingenieurwes (2007) 71: 47–58 57

Fig. 11 Flow coefficient versuspressure ratio

Cj :=

√

(1−ηs)+[

ωj ·ηs ·ln(

ηsηj

)

−(ωj −1)(ηs−ηj )

]

ωj

(

ηsηj

−1)

+1

Flow coefficient at each stepCcrit := max(C) Ccrit = 0.465

Maximum of flow coefficient

max := j ← 0while

∣∣Cj

∣∣ ≤ Ccrit ·0.9999

j ← j +1j

Step where maximum flow

coefficient occursηcrit := ηmax ηcrit = 0.691 critical pressure ratio

Flow coefficient versus pressure ratio (see Fig. 11)

Critical pressure ratio exceeds back pressure ratioData at critical pressure ratio are taken to calculate the flowcoefficient

η := ηmax η = 0.691 pressure ratioN := Nmax N = 0.034 boiling delay factorω := ωmax ω = 0.666 compressibility coefficientC := Cmax C = 0.465 flow coefficient

Estimation of the two-phase discharge coefficient of thesafety valve

ε := 1 − vl0

v0

[

ω(

ηsη

)

+1] ε = 0.2

void fraction in the throat area of the valveKd2ph := Kdg · ε+ (1 − ε) · Kdl Kd2ph = 0.554

derated two-phase discharge coefficient of the safety valve

mSV := Kd2ph ·C√

2p0v0

mSV = 1.055 ×104 kg/m2s

dischargeable mass flux through the safety valveASV = Qm

mSVASV = 6.581 ×10−4 m2

minimum required cross sectional area of the safety valve

dSV :=√

4π

ASV dSV = 28.9 mm

minimum required diameter of the safety valve

References

1. API 520 (2000) Sizing, selction, and installation of pressure-relieving devices in refineries, Part I sizing and selection, 7thEdition. American Petroleum Institute, January 2000

2. ISO 23521 (2006) Petroleum, petrochemical and natural gasindustries – Pressure-relieving and depressuring systems. DINDeutsches Institute fur Normung e.V., Beuth Verlag GmbH,Berlin

3. Leung JC (1986) A generalized correlation for one-componenthomogeneous equilibrium flashing choked flow. AIChE J 32(10):1743–1746

4. Leung JC (1990) Similarity between flashing and non-flashingtwo-phase flows. AIChE J 36(5):797–800

5. Fischer HG, Forrest JS, Grossel SS, Huff JE, Muller AR,Noronha JA, Shaw DA, Tilley BJ (1992) Emergency Re-lief System Design Using DIERS Technology, DIERS ProjectManual

6. Etchells J, Wilday J (1998) Workbook for chemical reactor reliefsystem sizing. HSE Contract Research Report 136

7. CCPS of AIChE (1998) Guidlines for pressure relief and effluenthandling systems. AIChE, New York

8. Bolle L, Downar-Zapolski P, Franco J, Seynhaeve JM (1995)Flashing water flow through a safety valve. J Loss Prev Proc8(2):111–126

9. Celata GP, Guidi G (1996) Problems about the sizing of two-phase flow safety valves. Heat Technol 14(1):67–95

10. Henry R, Fauske H (1971) The two-phase critical flow of one-component mixtures in nozzles, orifices, and short tubes. J HeatTransf 93(5):179–187

11. Diener R, Schmidt J (2004) Sizing of throttling devices for gasliquid two-phase flow, Part 1: safety valves. Process Saf Prog23(4):335–344

12. Diener R, Schmidt J (2005) Sizing of throttling devices for gasliquid two-phase flow, Part 2: control valves, orifices and nozzles.Process Saf Prog 24(1):29–37

13. Diener R, Schmidt J (1998) Extended ω-method applicable forlow inlet mass flow qualities. 13th Mtg ISO/TC185/WG1, Lud-wigshafen, Germany, 15–16 June 1998

1 3

58 Forsch Ingenieurwes (2007) 71: 47–58

14. Schmidt J, Friedel L, Westpahl F, Wilday J, Gruden M, van derGeld C (2001) Sizing of Safety Valves for Two Phase Gas/LiquidMixtures. 10th Int. Symposium on Loss Prevention and SafetyPromotion in the Process Industrie, Stockholm, 19–21 June 2001

15. ISO/DIS-4126-10 (2006) Safety devices for protection against ex-cessive pressure – sizing of safety valves and connected inlet andoutlet lines for gas/liquid two-phase flow. DIN Deutsches Institutefur Normung e.V., Beuth Verlag GmbH, Berlin

16. Diener R, Kiesbauer J, Schmidt J (2005) Improved valve sizingfor multiphase flow – HNE-DS method based on an expansionfactor similar to gaseous media to account for changes in mixturedensity. Hydrocarb Process 84(3):59–64

17. ISO 5167 2/3 (2000) Measurement of fluid flow by means ofpressure differential devices inserted in circular cross-sectionconduits running full. DIN Deutsches Institute fur Normung e.V.,Beuth Verlag GmbH, Berlin

18. EN-ISO 9300 (2003) Measurement of gas flow by means of crit-ical venturi nozzles. DIN Deutsches Institute fur Normung e.V.,Beuth Verlag GmbH, Berlin

19. Woodward JL (1995) An amended method for calculation omegafor a homogeneous equilibrium model of prediting dischargerates. J Loss Prev Proc 8(5):253–259

20. Leung JC (1994) Flashing flow discharge of initially sub-cooledliquid in pipes. J Fluid Eng 116:643–645

21. Leung JC (1988) A Generalized Correlation for Flashing ChokedFlow of Initially Sub-cooled Liquid. AIChE J 34(4):688–691

22. Sozzi GL, Sutherland WA (1975) Critical flows of saturated andsubcritical water at high pressure. General Electric, San Jose, CA,NEDO-13418, July 1975

23. Plesset MS, Zwick SA (1954) The groth of vapour bubble insuperheated liquids. J Appl Phys 25(4):493–500

24. Abuaf N, Jones OC, Wu BJC (1983) Critical flow in Nozzles withsub-cooled inlet conditions. J Heat Transf 105:379–383

25. Alamgir M, Lienhard JH (1981) Correlation of pressure under-shoot during hot-water depressurisation. Trans ASME 103:52–55

26. Boivin JY (1979) Two-phase critical flow in long nozzles. NuclTechnol 46

27. Lee DH, Swinnerton D (1983) Critical flow of subcooled water atvery high pressure relevant to an ATWS. Safety and EngineeringScience Division

28. Simoneau RJ, Hendricks RC (1984) Two phase flow of cryogenicfluids in converging-diverging nozzles. NASA Technical Paper

29. Veneau T (1995) Etude experimentale et modelisation de ladecompression d’un reservoir de stockage de propane. These dedoctorat – Institut National Polytechnique de Grenoble

30. Seynhaeve (2006) private communication31. Boccardi G, Bubbico R, Celata GP, Mazzarotta B (2005) Two-

phase flow through pressure safety valves. Experimental investi-gation and model prediction. Chem Eng Sci 60:5284–5293

32. Bolle L, Downar-Zapolski P, Franco J, Seynhaeve JM (1995)Flashing water flow through a safety valve. J Loss Prev Proc8(2):111–126

33. Sallet DW (1984) Thermal hydraulics of valves for nuclear appli-cations. Nucl Sci Eng 88:220–244

34. Lenzing T (2001) Theoretische und Experimentelle Untersuchun-gen zu dem uber Vollhubsicherheitsventile abfuhrbaren Massen-strom bei Einphasen- und Zweiphasenstromung, Fortschritt-Berichte VDI Reihe 3 Nr. 718 VDI Verlag

35. Darby R (2004) On two-phase frozen and flashing flows in safetyrelief valves – recommended calculation method and the properuse of the discharge coefficient. J Loss Prev Proc 17:255–259

36. Schmidt J (2006) Sizing of Safety Valves, Control Valves, Ori-fices and Nozzles – HNE-DS model applied for two-phase crit-ical flow with saturated and initially subcooled liquid. EuropeanDIERS User Group meeting, London, UK, April 2006

37. Kim SW, No HC (2001) Subcooled water critical pressure andcritical flow rate in safety valve. Int J Heat Mass Transf 44:4567–4577

1 3