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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…
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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 References 1. 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