water mixtures

ARTICLE IN PRESS Journal of Loss Prevention in the Process Industries 21 (2008) 567–578

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Computation of nozzle flow capacities for superheated steam, subcooled water, and saturated steam/water mixtures M.A. Westman Air Products and Chemicals, Inc., Allentown, PA, USA

a r t i c l e in fo

abstract

Article history: Received 27 March 2008 Accepted 12 June 2008

A new computational tool models the discharge of steam, water, and saturated steam–water mixtures from safety relief valve nozzles, employing thermodynamic-property formulas derived from the Helmholtz equation and programmed into spreadsheet-based software. The computational approach maps an isentropic process from initial conditions in a series of thermodynamic states at each of which the mass flux is computed. The user can identify the maximum mass flux from the generated mass-flux pressure graphical profile. The work describes other mass-flux predictive techniques commonly used in industry and then compares the results of the proposed method against those of the others. The data indicate a very high degree of correspondence between the proposed method and the Napier equation for saturated and superheated steam. The proposed method produces saturated-liquid results consistent with those of the other methods, particularly ASME VIII/1. For the relief of saturated liquid–vapor mixtures, the proposed method increasingly overpredicts maximum mass flux relative to the HEM-based o-method for increasing pressures and qualities. In the low-quality range, for which there is experimental data to which to compare, the proposed method severely underpredicts measured results involving flows out of nozzles of lengths less than ‘‘relaxation’’ length, but for flows out of longer nozzles, the underprediction is significantly less. The proposed method’s subcooled-water results compared to measured data also indicates large underpredictions for sub-relaxation lengths with improved results at generally greater degrees of subcooling at greater lengths. The proposed technique shows generally good agreement with other established predictive methods within demonstrated boundaries and has advantages in stand-alone capability and versatility. & 2008 Elsevier Ltd. All rights reserved.

Keywords: Nozzle Pressure relief Saturated Steam Subcooled water

The advantages of this algorithm include:

1. Introduction Water is among the most common fluids used in the process industries. Despite its pervasive use, the design of systems for the emergency relief of water under certain inlet conditions has been particularly challenging for many engineers who do not have access to computer programs that use sophisticated databases of thermodynamic properties. Now, using fundamentally derived equations and widely available spreadsheet software, engineers can write computer programs of considerable utility and accuracy to predict relief valve nozzle mass flow capacities and other information applicable to the proper design of safety relief systems. The objective of this paper is to describe a computational technique by which nozzle mass flow capacities can be predicted for the relief of superheated and dry saturated steam, subcooled water, and saturated steam/water mixtures, including 100% saturated water, and to compare this technique to other reported methods.

E-mail address: [email protected] 0950-4230/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.jlp.2008.06.002

 Ease in computing nozzle capacities for complex steam and

 



water relief systems without the need for elaborate and costly sizing software or databases of steam thermodynamic properties. Ability to analyze nozzle capacity resulting from various inlet thermodynamic conditions with a single tool. Improved ability through graphical output to predict phase conditions at the nozzle throat, enabling the user to better select safety relief valve trim and other features best suited for the specified relief conditions. Availability of a stand-alone, laptop-based analytical tool of high transparency and portability.

The proposed technique is most effective in predicting mass flow rates of systems presumed to behave on the basis of thermodynamic and mechanical equilibrium. However, it can also provide important information for the enhancement of nonequilibrium-based predictive techniques.

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t Z

Nomenclature A, C, E, F coefficients in water-property correlations T temperature G mass flux R gas constant (steam) t temperature h enthalpy u internal energy p pressure pset set pressure patm atmospheric pressure plim limiting pressure v specific volume s entropy K velocity heads of pressure loss Kd safety relief valve coefficient Ksh superheat correction factor Kn Napier-equation correction coefficient r density

Subscripts crit r lim 0 s sc sh f g t c ne tp

2. Thermodynamic basis The fundamental relationship from which the thermodynamic properties are derived is the Helmholtz free-energy equation as given in Keenan, Keyes, Hill, & Moore (1978):

cðr; TÞ ¼ c0 ðTÞ þ RT½ln r þ rQ ðr; tÞ

(1)

7 X ðt  taj Þj2 Q ðr; tÞ ¼ ðt  tc Þ

Aij ðr  raj Þi1 þ eEr

i¼1

10 X

#



qðctÞ qt

(10)

r

(11)

(12)

i¼9

6 X Ci

ti1 i¼1

þ C 7 ln T þ C 8

ln T

(3)

t

 ¼

7 X

r

Z j ðt  taj Þj2 þ ðt  tc Þ

j¼1

7 X

Z j ðj  2Þðt  taj Þj3

(4)

j¼1

and with respect to density at constant temperature,

qQ qr



(2)

Eq. (2) is differentiated with respect to temperature at constant density,



u ¼ r2

(9)

t

3. Computational approach

c0 ðtÞ ¼

qQ qt

qr

h ¼ u þ pv

Aij ri9

and



These results are used to formulate   qc p ¼ r2

The enthalpy then can be calculated via

j¼1 8 X

relating to critical temperature or pressure relating to reduced temperature or pressure limiting pressure conditions inlet or stagnation saturated conditions subcooled conditions superheated conditions liquid phase gas or vapor phase conditions at nozzle throat sonically limited critical conditions nonequilibrium conditions two-phase

  qc s¼ qT r

in which

"

inverse temperature, 1000/T pressure ratio p/p0



¼ ðt  tc Þ

t

7 X dZ j ðt  taj Þj2 dr j¼1

(5)

in which Z j ðrÞ ¼

8 X

Aij ðr  raj Þi1 þ eEr

i¼1

10 X

Aij ri9

(6)

i¼9

Also, 6 dc0 X ði  1ÞC i T i2 C 7 C8 ¼ þ ð1 þ ln TÞ þ i1 dT T 1000 ð1000Þ i¼1

d ðtc0 Þ ¼ dt

6 X ð2  iÞC i i¼1

ti1

(7)

  1000 C8 þ C 7 ln 1 

t

t

(8)

The proposed method presumes a quasi-static thermodynamic-equilibrium process in which maximum mass flux is sought through a series of isentropic-flash calculations. With the availability of thermodynamic quantities as functions of density and temperature, an initial thermodynamic state can be established and from it an isentropic process mapped in a series of decreasing pressures. For initial subcooled-liquid or superheated-steam conditions, process specifications call out the initial nozzle-inlet (stagnation) pressure p0 and temperature T0 that establish the degree of superheat or subcooling. Given the initial process conditions, the stagnation density r0 is determined implicitly from Eq. (9). Once r0 is calculated, the other stagnation thermodynamic properties, including the initial entropy maintained as a baseline for every calculation associated with a decreasing pressure down the isentropic path, are computed explicitly. Then all thermodynamic properties are computed from initial (stagnation) conditions on an isentropic-flash path to the saturation curve, defined by " # 8 X ps ¼ exp t  105 ðt crit  tÞ F i ð0:65  0:01tÞi1 (13) pcrit i¼1 To find the intercept of the isentropic path and the saturation curve at ps, a solver tool is used to vary r and T to solve Eq. (13) while using Ds ¼ 0 as a constraint. Once this is done, the user may designate point j on the saturation curve and select a suitable

ARTICLE IN PRESS M.A. Westman / Journal of Loss Prevention in the Process Industries 21 (2008) 567–578

Initial subcooled-water point 0 [0,T0] sc

Initial superheatedsteam

Saturation curve

Pressure

ps,x = 0

Critical point ps,x = 1

s = 0 pt

569

A typical graphical profile of G against pressure is shown in Fig. 2, from which Gc can be readily identified. For isentropic flows involving high-subcool initial conditions, dG/dp40 continues along the flash path to the saturation curve, where the throat pressure is limited to ps, x ¼ 0, and dGc/dp|t ¼ 0 is not reached. The difference between h0 and h(ps, x ¼ 0) is used to compute the subcooled-liquid mass flux Gsc through Eq. (16). This method corresponds to more common subcooled mass-flux calculations that devolve from Eq. (15), such as (Fisher et al., 1992) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2½p0  ps ðT 0 Þ Gsc ¼ vf 0

Initial saturated point 0 [0,T0] sc

(18)

Gc

Enthalpy Fig. 1. Isentropic paths on the pressure-enthalpy diagram.

series of computation points i ¼ 1; 2; . . . ; j between the initial conditions and the saturation intercept, then run the flash routine to find values of r and T that satisfy Eq. (9) against the isentropic constraint at each point. The computation of the other intermediate thermodynamic properties follows from these results. The mapping of the isentropic process continues below the saturation curve in a series of flash points i ¼ j þ 1; j þ 2; . . . ; n corresponding to a series of decreasing saturation pressures with their attendant vapor qualities, xi ¼

s0  sfi sgi  sfi

(14)

For saturated initial conditions, process specifications call out the initial nozzle-inlet (stagnation) conditions [p0,T0]s and the initial quality x0. The properties of saturated vapor and liquid are obtained and correlated vapor qualities computed on the isentropic path ending at a convenient pressure. Fig. 1 shows typical conceptual isentropic processes starting from arbitrary subcooled, superheated, and saturated initial conditions. The mass flux G is computed for a series of flash points [pi, Ti] from initial conditions [p0, T0] on the isentropic path through the use of the Bernoulli equation (Darby, Self, & Edwards, 2001), qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R 0 v dp (15) G¼ v

G

p 0 p1…

pc

pn-1 pn

Pressure Fig. 2. Mass-flux vs. pressure, saturated steam, saturated steam–water mixtures, and high-superheat steam.

Gsc

G

or the corresponding expression from the First Law of Thermodynamics, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ðh0  hÞ G¼ (16) v The solutions of enthalpy and density are so precise that Eq. (16) may be used for the computation of G at all flash points, eliminating the need for numerical integration of Eq. (15). Therefore, with the resulting array of data, the user can calculate values of G for all computation points in ptppipp0 using the enthalpy-difference form of Eq. (16) and easily identify the critical mass flux where dG/dp is limited.

pi

p0 p 1…

pi

ps, x = 0

pn-1 pn

Fig. 3. Mass flux vs. pressure, high-subcooled water.

Gc

G 4. Mass-flux vs. pressure profiles For saturated and high-superheated flows, a sonically limited maximum of dG/dp is reached, i.e.  dGc  ¼0 (17) dp t

p 0 p1… pi

ps, x = 0

pc

pn-1 pn

Fig. 4. Mass flux vs. pressure, low-subcooled water and low-superheat steam.

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in general keeping with the isentropic process in which neither T0 nor vf0 varies significantly. A typical graphical identification of high-subcooled Gsc is shown in Fig. 3. However, for low-subcool initial conditions, dG/dp40 prevails even as the flash proceeds below the saturation curve, increasing to dGc/dp|t ¼ 0 at a sonically limited pressure pc at the nozzle throat with a corresponding quality xt40, indicating the attainment of liquid flashing in the nozzle. A typical graphical depiction is represented in Fig. 4. For low-superheat steam initial conditions, pc is not attained in the superheated region, but dG/dp40 continues below the saturation line, increasing to dGc/dp|t ¼ 0 at a sonically limited pressure pc at the nozzle throat with a corresponding quality xt40. A typical graphical depiction is similar to the low-subcool process and is also represented in Fig. 4.

treats flashing flows as though they are single-phase fluids in mechanical and thermal equilibrium, yielding conservative capacities for safety relief valve nozzles of lengths greater than 10 cm. A simplified version of HEM (Leung, 1986), commonly known as the o-method, uses a general equation of state for two-phase expansion,   v p (21) ¼o 0 1 þ1 vf 0 p for which v ¼ xvg+(1x)vf and !2 xv C T 0 p0 vfg0 o ¼ fg0 þ pf 0 v0 vf 0 hfg0 with the critical-pressure ratio Zc found implicitly from

Z2c þ ðo2  2oÞð1  Zc Þ2 þ 2o2 ln o þ 2o2 ð1  Zc Þ ¼ 0 Then the critical flow rate can be found by sffiffiffiffiffiffiffiffiffiffiffi p0 Gc ¼ Zc ovf 0

5. Other reported methods 5.1. Napier equation The Napier equation, the basic form of which is Gc ¼ 0:1459K sh K n p0

(19) 2

(with Gc in units of kg/s cm and p0 in MPa-a) was developed through experimentation with the relatively low saturated-steam pressures available at the time and meant to serve as a simplified mass flux calculation in initial-pressure range of 0 MPa-aop0p 10.3 MPa-a. As a linear function it provides a good approximation of the isentropic nonlinear process in that range, but in 10.3 MPaaop0p22.1 MPa-a it underpredicts the increasingly nonlinear isentropic-method mass flux, and an aligning correction factor Kn must be applied (p0 in MPa-a): 8 0 Mpa  aoP0 p10:3 MPa  a > < 1:0 K n ¼ 27:6442p0  1000 > : 33:2427p  1061 10:3 Mpa  aoP 0 p22:1 MPa  a 0 (20) The superheat correction factor Ksh is commonly available in tabular form and applied according to the initial pressure and temperature. As reported by Thompson and Buxton (1979), theoretical maximum isentropic-flow results for saturated initial conditions underpredict those of the Napier equation by as much as 3% for pressures between 0 and 10.3 MPa-a. Thompson and Buxton observed that the nonequilibrium effect of supersaturation can prevail in saturated-steam flow through comparatively short nozzles. Due to the short transit times, the high-velocity steam does not achieve thermodynamic equilibrium at the throat but instead reaches a state of metastable nonequilibrium through the nozzle. Their experiments in the initialpressure range of 2.8–15.9 MPa-a, in which a particular design of safety relief valve nozzle was used, confirmed the greater mass flux that attends the supersaturation effect against the Napier equation’s prediction under those conditions. This work will explore supersaturation further when it discusses the Homogeneous Nonequilibrium Method as applied to the prediction of Gc for saturated-liquid and subcooled-liquid initial conditions.

(22)

(23)

(24)

A further refinement of the o-method permits analysis of nozzle flows for subcooled initial conditions by means of another generalized correlation (Leung & Grolmes, 1988). In place of Eq. (21), this method employs   v p (25) ¼o s 1 þ1 vf 0 p with the modified o parameter, !2 C pf 0 T 0 ps vfg0 o¼ vf 0 hfg0

(26)

in which all the physical properties are taken at the point on the saturation curve corresponding to T0. The critical-pressure ratio Zc is then found implicitly from     o þ ð1=oÞ  2 2 Z Zc  2Zc ðo  1Þ þ oZs ln c 2Zs Zs 3 þ oZs  1 ¼ 0 (27) 2 and the critical mass flux from sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p0 Gc ¼ Zc ovf 0 Zs

(28)

Expanding the logarithmic term in Eq. (27) in a Taylor series yields the explicit approximate solution to that equation, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  "  # 2o 1 2o  1 1 1 Zc ¼ Zs (29) 2o 2o  1 Zs which to produce real results must satisfy

Zs X

2o  1 2o

(30)

This inequality delimits the boundary between the ‘‘lowsubcool’’ region, in which the inequality is satisfied and critical flow will result from flashing in the nozzle, and the ‘‘highsubcool’’ region, in which flashing will not occur in the nozzle, and all-liquid flow will result according to Eq. (18). 5.3. Homogeneous Nonequilibrium (HNE) method

5.2. Homogeneous Equilibrium (HEM) o-method The Homogeneous Equilibrium Method (HEM) has gained wide acceptance as a generally conservative means of calculating saturated flashing-flow mass-flux capacities (Huff, 1985). HEM

Homogeneous flashing nozzle-flow models assume thermodynamic and mechanical equilibrium. Homogeneity requires that both phases flow at the same velocity (no slip), and that heat, mass, and momentum be transferred adiabatically between the

ARTICLE IN PRESS M.A. Westman / Journal of Loss Prevention in the Process Industries 21 (2008) 567–578

two phases according to the local temperature and pressure. Therefore, flashing would occur when ps is reached. However, in short nozzles of lengths of less than an experimentally observed ‘‘relaxation length’’ Le of approximately 10 cm, the residence time is too short to permit the onset of bubble nucleation, and the liquid component undergoes a delay in flashing (Darby et al., 2001). To account for these phenomena, Henry and Fauske (1971) proposed the homogenous nonequilibrium (HNE) method. Fauske (1985, 1999) also developed a simplified, explicit no-slip approach based on stagnation conditions in which, for LoLe, vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 u2½p  pðT 0 Þ  hfg0 ðT 0 Þs s Gx0 ¼0 ¼ t 0 (31) þ 2 vf 0 vfg ðT 0 Þs T o C pf 0 Nne 0

2

Nne ¼

vf 0 hfg0 ðT 0 Þs 2ðp0  plim Þv2fg ðT 0 Þs T 0 C pf 0 0

þ

L Le

(32)

and, for LXLe, for which Nne ¼ 1 and frictional effects are presumed to non-negligible along the nozzle length, vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2 u2½p  pðT 0 Þ  hfg 0 ðT 0 Þs s u 0 þ 2 u vf 0 vfg ðT 0 Þs T o C pf 0 t 0 (33) Gx0 ¼0 ¼ 1þK The limiting pressure plim in Eq. (32) is the critical pressure for flashing flows resulting from saturated or low-subcooled initial conditions (Figs. 2 and 4, respectively) or the saturation pressure resulting from high-subcool initial conditions (Fig. 3). Eqs. (31) and (33) account for subcooled initial conditions in the first term under the radical, which goes to zero at saturated initial conditions, and in evaluating the latent heat and specificvolume differences at saturation conditions corresponding to T0. A conservative approximation of saturated-liquid initial conditions (x0 ¼ 0) is obtained when the frictional nozzle loss is ignored and Eq. (31) reduces to the relationship commonly known as the Equilibrium Rate Model (ERM): Gx0 ¼0 ¼

hfg0 qffiffiffiffiffiffiffiffiffiffiffiffiffi ffi vfg0 T o C pf 0

(34)

With its close relationship to Eqs. (22) and (24), ERM has been used as an approximate method for computing maximum saturated-liquid flow, and its results will be compared to those of the other methods for that application later in this work. For mixed liquid-vapor flashing flows, the overall two-phase mass flux is computed by proportioning the independently calculated mass fluxes of the liquid and vapor phases, !1=2 1  x0 x0 Gtp ¼ þ 2 (35) G2x0 ¼0 Gx0 ¼1 in which the mass flux corresponding to x0 ¼ 0 is Eqs. (31), (33) or (34), and the mass flux corresponding to x0 ¼ 1.0 is the Napier equation. (Although other works discussing saturated mixtures generally employ the isentropic ideal-gas sonic-limit capacity equation in this term, this work, in investigating water exclusively, will use Eq. (19) instead.) 5.4. ASME Section VIII, Division 1, Appendix 11 ASME Mandatory Appendix 11, Capacity Conversions for Safety Valves (2004) contains Fig. 11-2M, a graph of saturated-water capacity based on an isentropic equilibrium calculation method that the Code does not identify. The Code allows the use of this graph for the sizing of safety relief valve models of contoured

571

nozzle construction with throat-to-inlet diameter ratios of 0.25–0.80 and valve coefficients Kd40.90. 6. Safety relief valve coefficient Kd The safety relief valve coefficient of discharge Kd is a factor applied to mass-flux calculations to account for a nozzle’s variation from ideal behavior. For safety relief valves designed for the discharge of steam, it is the ratio of measured nozzle flow of dry saturated steam to theoretical flow computed from the Napier equation. While there is no consensus in industry on Kd values applicable to two-phase flashing flow, work done by Darby, Meiller, and Stockton (2000, 2001) suggests that for flashing flows, whether the stagnation condition is superheated, saturated, or sufficiently low-subcooled, Kdffi1.0, while for high-subcooled or nonequilibrium supersaturated flows in which the liquid phase is maintained through the nozzle, Kdffi0.7. Saturated-mixture G calculations involving treatment of particular phases in separate terms, such as Eq. (35), can be corrected by a single Kd or weighted according to the vapor or liquid Kd most commonly used with each term’s associated phase. While the study of multiphase Kd is an important ongoing effort in the area of safety relief technology, this work will presume that the nature of homogeneous flow implies the application of a single constant or variable Kd factor to any calculated two-phase G. Therefore, for clarity and ease of comparison, the calculated mass fluxes presented in this work will be unadjusted for Kd. 7. Comparison of results 7.1. Saturated and superheated steam Table 1 compares the results of the proposed isentropic enthalpy-difference method and the Napier equation for inlet pressures between 1 and 21 MPa-a. The results of the two techniques were highly similar with the isentropic technique underpredicting the Napier equation by approximately 3% in the vicinity of 4.0–6.0 MPa-a as Thompson and Buxton reported. An overprediction of approximately the same magnitude develops in the upper end of the pressure range nearer to pcrit (22.088 MPa-a). Table 2 compares results for selected pressures at saturation and corresponding arrays of selected superheat temperatures. Here the superheat correlations are seen to be generally closer than even those for the corresponding saturation, especially for superheat temperatures greater than 10% of saturation in absolute terms. 7.2. Saturated liquid and saturated vapor–liquid mixtures A comparison of the proposed method, o-method, ERM, and ASME VIII-1 Fig. 11-2M for saturated liquid are shown in Table 3 and graphically in Fig. 5 for inlet pressures between 1.0 and 21.0 MPa-a. Mass flux is plotted against gauge set pressures pset that correspond to absolute inlet stagnation pressures p0 ¼ 1.1pset+patm as the Code permits in the employment of single safety relief valves. A large difference exists between the two isentropic methods at the lower end of the pressure range, with the proposed technique exceeding Code-provided values by nearly 50% at 1.0 MPa-a. This margin tapers off at greater pressures, but an overprediction in excess of 4% still exists at 21.0 MPa-a. The o-method results correlate closely with those of the proposed technique through midrange, but then, as Leung predicted for

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Table 1 Comparison of proposed method and corrected Napier equation for saturated steam Initial saturation pressure (MPa-a)

Saturation temperature (C)

Isentropic model mass flux GcIsen (kg/s cm2)

Corrected Napier equation GcNapier (kg/s cm2)

(GcIsenGcNapier)/ GcNapier (%)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

179.9093 212.4184 233.9008 250.4050 263.9933 275.6391 285.8850 295.0670 303.4083 311.0652 318.1516 324.7531 330.9355 336.7509 342.2409 347.4393 352.3741 357.0687 361.5428 365.8134 369.8950

0.1442 0.2845 0.4245 0.5655 0.7080 0.8518 0.9980 1.1465 1.2969 1.4514 1.6073 1.7679 1.9350 2.1050 2.2803 2.4647 2.6595 2.8650 3.0837 3.3358 3.6374

0.1459 0.2918 0.4376 0.5835 0.7294 0.8753 1.0211 1.1670 1.3129 1.4588 1.6060 1.7669 1.9319 2.1019 2.2776 2.4601 2.6508 2.8515 3.0645 3.2929 3.5409

1.15 2.50 3.00 3.08 2.94 2.68 2.27 1.76 1.22 0.51 0.08 0.06 0.16 0.15 0.12 0.19 0.33 0.47 0.63 1.30 2.72

Table 2 Comparison of proposed method and corrected Napier equation for superheated steam Pressure (MPa-a)

1

3

5

10

15

21

Saturation temperature (C)

Superheat temperature (C)

Superheat correction factor

Isentropic model mass flux GcIsen (kg/s cm2)

Corrected Napier-equation mass flux GcNapier (kg/s cm2)

(GcIsenGcNapier)/ GcNapier (%)

207.6940 267.6940 432.6940 547.6940

0.9871 0.9331 0.8046 0.7429

0.1442 0.1418 0.1359 0.1172 0.1081

0.1459 0.1440 0.1361 0.1174 0.1084

1.15 1.51 0.19 0.11 0.24

261.6694 321.6694 486.6694 601.6694

0.9602 0.9024 0.7801 0.7209

0.4245 0.4163 0.3951 0.3409 0.3149

0.4376 0.4202 0.3949 0.3414 0.3155

3.00 0.94 0.04 0.16 0.18

291.7550 351.7550 516.7550 631.7550

0.9707 0.8925 0.7600 0.7123

0.7080 0.6929 0.6510 0.5598 0.5172

0.7294 0.7080 0.6510 0.5543 0.5195

2.94 2.13 0.00 0.98 0.46

378.8083 428.8083 478.8083 528.8083

0.9131 0.8581 0.8131 0.7781

1.4514 1.3287 1.2461 1.1827 1.1305

1.4588 1.3320 1.2518 1.1861 1.1351

0.51 0.25 0.45 0.29 0.40

409.9697 459.9697 509.9697 559.9697

0.8807 0.8212 0.7762 0.7423

2.2803 2.0093 1.8681 1.7652 1.6826

2.2776 2.0059 1.8703 1.7679 1.6906

0.12 0.17 0.12 0.15 0.48

437.6587 487.6587 537.6587 587.6587

0.8056 0.7405 0.6955 0.6675

3.6374 2.8599 2.6295 2.4703 2.3463

3.5409 2.8526 2.6221 2.4627 2.3636

2.72 0.26 0.28 0.31 0.73

179.9093

233.9008

263.9933

311.0652

342.2409

369.8950

reduced pressures pr0 exceeding about 0.5 and reduced temperatures Tr0 exceeding about 0.9, diverge sharply in the upper part of the range. Leung notes that the o-method results can be

improved using Tr0 as an additional parameter and substituting dhf/dT|0 in place of Cpf0 in Eq. (22). The curve of the approximate ERM formula exceeds the other methods by an

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573

Table 3 Comparison of proposed method, ASME VIII-1 Fig. 11-2M, and omega method for saturated water Set pressure (MPa-g)

Absolute pressure (based on 10% accumulation) (MPa-a)

Isentropic model ASME VIII/1 mass flux GcIsen mass flux GcASME (kg/s cm2) (kg/s cm2)

HEM model mass flux GcHEM (kg/s cm2)

ERM model mass flux GcERM (kg/s cm2)

(GcIsenGcASME)/ GcASME (%)

(GcHEMGcASME)/ GcASME (%)

(GERMGcASME)/ GcASME (%)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

1.2013 2.3013 3.4013 4.5013 5.6013 6.7013 7.8013 8.9013 10.0013 11.1013 12.2013 13.3013 14.4013 15.5013 16.6013 17.7013 18.8013 19.9013 21.0013

0.7381 1.2042 1.6013 1.9532 2.2746 2.5693 2.8423 3.0983 3.3368 3.5603 3.7698 3.9677 4.1519 4.3214 4.4768 4.6134 4.7276 4.8065 4.8264

0.7492 1.2168 1.6099 1.9545 2.2622 2.5395 2.7904 3.0164 3.2188 3.3972 3.5504 3.6753 3.7687 3.8237 3.8297 3.7697 3.6144 3.3036 2.6657

0.8566 1.4429 1.9573 2.4213 2.8441 3.2310 3.5846 3.9062 4.1958 4.4518 4.6712 4.8491 4.9776 5.0446 5.0316 4.9088 4.6250 4.0824 3.0461

47.61 35.48 25.32 17.19 16.98 15.62 13.69 11.54 9.20 9.55 10.33 9.87 9.90 9.56 8.90 8.55 8.40 6.16 4.35

49.85 36.89 25.99 17.27 16.34 14.28 11.62 8.59 5.34 4.53 3.91 1.78 0.24 3.06 6.84 11.30 17.12 27.04 42.36

71.32 62.32 53.18 45.28 46.27 45.39 43.38 40.62 37.32 36.98 36.72 34.28 31.76 27.89 22.39 15.50 6.05 9.84 34.14

0.50 0.89 1.28 1.67 1.94 2.22 2.50 2.78 3.06 3.25 3.42 3.61 3.78 3.94 4.11 4.25 4.36 4.53 4.63

Isentropic Enthalpy

Table 5 presents the results of the proposed technique,

Omega Model

Difference ASME VIII/1

o-method, and simplified-HNE method against data on flow

ERM Model

6.0000

5.0000

4.0000

3.0000

2.0000

1.0000

0.0000 1

3

5

7

9

11

13

15

17

19

Fig. 5. Comparison of proposed method, omega method, and ASME VIII-1 Fig. 112M for saturated liquid.

increasingly wider margin until its mass flux reaches a maximum at 15.5 Mpa-a and the solutions converge toward those of omethod as expected. Leung and Nazario (1990) reported agreement of within 10% throughout the pressure range between Fig. 11-2M and the basic HEM of which the o-method is an extension. Table 4 compares the proposed method and o-method for steam qualities of 0%, 10%, 50%, 75%, and 100% for selected saturation pressures. These data are consistent with the 0%quality results of Table 3 for the two techniques, showing a close correlation between the two methods at lower pressures up to 5.0 MPa-a but wider differences at 10.0 MPa-a and greater.

through 12.7-mm-diameter, rounded-entrance nozzles of highpressure (between 6.0 and 7.0 MPa-a), low-quality saturated mixtures collected by Sozzi and Sutherland (1975) as reported by Ilic, Banerjee, and Behling (1986). These data include measured mass-flux values for nozzle lengths of 0–228.6 mm so that the HNE-method results for both LoLe and LXLe can be compared to those of the equilibrium-based methods. The nozzle lengths chosen for this examination are representative of those of standard nozzles in commercially available safety relief valves. In employing Eq. (32) for LoLe with saturated inlet conditions, plim is set equal to the critical pressure pc resulting from liquid flashing at the nozzle throat in the absence of supersaturation. Values for pc are not available by means of the HNE method itself but must be found from other sources; in this work, pc are extracted from the results of the proposed method and employed in obtaining the simplified-HNE results. Here the results of the isentropic enthalpy-difference method and o-method are highly similar as they are in the previously discussed results irrespective of nozzle length; however, the simplified-HNE results show the greatest correlation to the measured data in LoLe, with underpredictions of measured values highest for flows of lower qualities but in good agreement for qualities above about 0.3%. For LXLe, the results of the three methods are in significantly closer agreement, with the proposed method and o-method underpredicting measured values on an average of about 10% greater than those of HNE for Lffi11 cm and improving to 3% greater for Lffi23 cm. For all nozzle lengths, the underpredictions of the proposed method and o-method from those of the HNE method can be broken out into two categories roughly corresponding to the quality ranges 0px0p0.002 and x040.002. For every nozzle length, in each of these two quality ranges, the differences between the proposed and HNE techniques, expressed as a fraction of the measured mass flux, C¼

GcIsen  GcHNE GcMeas

(36)

are close in value, suggesting an alignment of results of the two techniques based on p0, L, and x0.

ARTICLE IN PRESS 574

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Table 4 Comparison of proposed and omega methods for saturated liquid-vapor mixtures Pressure (Mpaa)

Saturation temperature (C)

x0

Isentropic model mass flux GcIsen (kg/s cm2)

HEM model mass flux GcHEM (kg/s cm2)

(GcIsenGcHEM)/GcHEM (%)

1

179.9093

1.00 0.75 0.50 0.10 0.00

0.1442 0.1647 0.1972 0.3615 0.6410

0.1340 0.1534 0.1847 0.3514 0.6511

7.62 7.38 6.76 2.89 1.55

3

233.8917

1.00 0.75 0.50 0.10 0.00

0.4245 0.4817 0.5705 0.9775 1.4621

0.3912 0.4458 0.5323 0.9495 1.4731

8.50 8.05 7.18 2.95 0.75

5

263.9772

1.00 0.75 0.50 0.10 0.00

0.7080 0.7992 0.9393 1.5328 2.1027

0.6496 0.7372 0.8739 1.4839 2.0980

8.98 8.42 7.49 3.30 0.23

10

311.0305

1.00 0.75 0.50 0.10 0.00

1.4514 1.6199 1.8670 2.7524 3.3363

1.3174 1.4784 1.7184 2.6118 3.2185

10.18 9.57 8.64 5.39 3.66

15

342.1919

1.00 0.75 0.50 0.10 0.00

2.2803 2.5094 2.8258 3.7782 4.2455

2.0143 2.2216 2.5106 3.3815 3.8040

13.21 12.95 12.56 11.73 11.60

21

369.8809

1.00 0.75 0.50 0.10 0.00

3.6374 3.8478 4.0996 4.6458 4.8246

2.2503 2.3337 2.4283 2.6122 2.6667

61.64 64.88 68.83 77.85 80.92

7.3. Subcooled liquid Table 6 compares the proposed method and the o-method subcool variant for both saturated and subcooled inlet conditions at selected pressures. Each degree of initial subcooling is depicted as a negative quality computed via xo ¼

v0  vf 0 ðP0 Þs vg0 ðP 0 Þs  vf 0 ðP 0 Þs

(37)

As with the saturated-water comparison, agreement is excellent in pr0o0.5, but as inlet pressures approach pcrit, the proposed method results exceed those of o-method by wide margins. Table 7 examines the results of the proposed technique, o-method, and simplified-HNE method against Sozzi and Sutherland’s (1975) measured mass fluxes (as reported by Ilic et al., 1986) of subcooled liquid of stagnation pressures between 6.0 and 7.0 MPa-a and inlet qualities between 0.0003 and 0.0024 through the 12.7-mm-diameter rounded-entrance nozzles of lengths 0–228.6 mm. Here also, the simplified-HNE results were markedly closer to the measured results, with the results of all three techniques varying most at qualities closer to zero. However, for L4Le, where equilibrium conditions are presumed to prevail, the proposed technique’s and o-method’s results improve against measured values. Similar to the results of the saturated-flow analyses from the Sozzi and Sutherland data, there appears to be two approximate saturation ranges, 0.0005Xx0X0.0012 and 0.00124x0, within which, respectively, for every nozzle length, all C are close in value (with the notable exception of L ¼ 0 cm, for

which all C appear nearly equal in 0.0005Xx0X0.0020), though with wider variations from the mean than those of the saturated results. In the HNE-method calculations for LoLe, plim is set at either pc for low-subcool initial conditions or ps,x ¼ 0 for high-subcool initial conditions; both the identification of type of subcooling and the determination of all required pc or ps,x ¼ 0 are provided in the proposed method output. Table 8 shows the results of the three techniques as applied to subcooled-liquid data presented by Bolle, Downar-Zapolski, Franco, and Seynhaeve (1996). Sixteen mass-flux measurements were recorded from nozzle flows of stagnation pressures between 0.405 and 0.614 MPa-a and inlet qualities between 0.00001 and 0.00013. A nozzle (Crosbys safety relief valve model 1D2 JLTJOS-15-A) of length 1.039 cm and diameter 1.039 cm was used in the HNE calculations. The simplified-HNE results once again offer closer predictions of measured results, with the results of all three techniques underpredicting the measured values most at qualities closer to zero, but for x0p0.00005 the techniques produce substantially identical results. The predictions of the proposed method and o-method align very closely with the HEM results that Bolle et al. (1996) reported. In 0.00001Xx0X0.00004 the average C is 0.1426, and in the range 0.000044x0X0.00013 the average C is 0.0270.

8. Conclusions Data from the proposed method indicate excellent agreement with the modified Napier equation for both saturated and

Table 5 Comparison of proposed, omega, and HNE methods for saturated liquid-vapor mixtures (Sozzi & Sutherland data) Initial quality x0

Measured mass flux GcMeas (kg/s cm2)

Isentropic model mass flux GcIsen (kg/s cm2)

HEM model mass flux GcHEM (kg/s cm2)

HNE model mass flux GcHNE, PLim ¼ Pc (kg/s cm2)

(GcIsenGcMeas)/ GcMeas (%)

(GcHEMGcMeas)/ GcMeas (%)

(GcHNEGcMeas)/ GcMeas (%)

C ¼ (GcIsenGcHNE)/ GcMeas

0.0 0.0 0.0 0.0 0.0 0.0

6.3431 5.7226 6.5844 5.9984 6.4810 6.1983

279.2924 272.5671 281.7723 275.6215 280.7181 277.7696

0.00000 0.00220 0.00430 0.00530 0.00600 0.00650

6.1031 4.5993 4.5607 4.1667 4.2859 4.1540

2.4760 2.2860 2.4981 2.3312 2.4548 2.3753

2.4525 2.2702 2.4670 2.3100 2.4249 2.3490

4.3777 4.0480 4.4097 4.0323 4.2417 4.1928

59.43 50.30 45.23 44.05 42.72 42.82

59.82 50.64 45.91 44.56 43.42 43.45

28.27 11.99 3.31 3.23 1.03 0.93

0.3116 0.3831 0.4192 0.4083 0.4169 0.4375

38.1 38.1 38.1 38.1 38.1 38.1

6.2880 6.3155 6.0260 6.8533 6.7016 6.1983

278.7155 279.0044 275.9210 284.4553 282.9519 277.7696

0.00000 0.00100 0.00150 0.00270 0.00370 0.00400

4.3762 3.8669 3.5398 3.1936 3.1936 2.9686

2.4609 2.4582 2.3764 2.5826 2.5330 2.3986

2.4388 2.4346 2.3563 2.5479 2.5015 2.3741

3.2703 3.2676 3.1931 3.4719 3.3506 3.2268

43.77 36.43 32.87 19.13 20.69 19.20

44.27 37.04 33.43 20.22 21.67 20.02

25.27 15.50 9.80 8.71 4.91 8.70

0.1850 0.2093 0.2307 0.2785 0.2560 0.2790

63.5 63.5 63.5 63.5

6.8120 6.6672 6.8671 6.7085

284.0478 282.6066 284.5907 283.0208

0.00230 0.00320 0.00370 0.00450

3.1492 3.0418 2.9881 2.9881

2.5745 2.5293 2.5758 2.5278

2.5420 2.4983 2.5411 2.4949

3.0119 3.0174 3.0194 3.0198

18.25 16.85 13.80 15.41

19.28 17.87 14.96 16.51

4.36 0.80 1.05 1.06

0.1389 0.1605 0.1485 0.1647

114.3 114.3 114.3 114.3 114.3

6.2052 5.7640 6.7568 6.6327 6.1501

277.8428 273.0323 283.5016 282.2600 277.2560

0.00080 0.00110 0.00270 0.00300 0.00330

3.1248 2.9783 2.7342 2.7342 2.6610

2.4316 2.3083 2.5579 2.5217 2.3913

2.4093 2.2929 2.5247 2.4917 2.3693

2.7309 2.5868 2.8785 2.8378 2.6856

22.18 22.50 6.45 7.77 10.13

22.90 23.02 7.66 8.87 10.96

12.60 13.14 5.28 3.79 0.93

0.0958 0.0935 0.1172 0.1156 0.1106

190.5 190.5 190.5 190.5 190.5

6.3776 6.8258 6.5431 6.8120 6.5017

279.6510 284.1838 281.3522 284.0478 280.9300

0.00000 0.00220 0.00300 0.00440 0.00490

3.1150 2.8231 2.6795 2.6224 2.5150

2.4853 2.5803 2.4995 2.5562 2.4702

2.4607 2.5465 2.4700 2.5210 2.4410

2.6254 2.7327 2.6453 2.7088 2.6163

20.22 8.60 6.72 2.52 1.78

21.01 9.80 7.82 3.87 2.94

15.72 3.20 1.28 3.30 4.03

0.0450 0.0540 0.0544 0.0582 0.0581

228.6 228.6 228.6 228.6 228.6 228.6 228.6

6.5500 6.1708 6.9223 6.7844 6.2466 6.8878 6.5293

281.4223 277.4765 285.1302 283.7751 278.2802 284.7934 281.2117

0.00050 0.00100 0.00200 0.00300 0.00370 0.00450 0.00480

3.1492 2.7098 2.6854 2.5877 2.4266 2.5145 2.4413

2.5250 2.4191 2.6062 2.5619 2.4144 2.5747 2.4790

2.4978 2.3984 2.5713 2.5286 2.3894 2.5380 2.4483

2.5959 2.4845 2.6853 2.6395 2.4836 2.6552 2.5542

19.82 10.73 2.95 1.00 0.50 2.39 1.55

20.68 11.49 4.25 2.29 1.53 0.93 0.29

17.57 8.31 0.00 2.00 2.35 5.60 4.63

0.0225 0.0241 0.0294 0.0300 0.0285 0.0320 0.0308

ARTICLE IN PRESS

Temperature T0 (C)

M.A. Westman / Journal of Loss Prevention in the Process Industries 21 (2008) 567–578

Stagnation pressure P0 (MPaa)

Nozzle length (mm)

575

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Table 6 Comparison of proposed and omega methods for subcooled water Pressure (Mpa-a)

0.5000

0.6250

0.7500

0.8750

1

3

5

10

15

21

Saturation temperature (C)

Subcooled temperature (C)

Initial quality x0

Isentropic model mass HEM model mass flux flux GcIsen (kg/s cm2) GcHEM (kg/s cm2)

(GcIsenGcHEM)/GcHEM (%)

148.5249 114.4040

0.00000 0.00001 0.00010

0.3722 0.8862 2.5176

0.3785 0.8867 2.5169

1.65 0.05 0.03

157.9191 132.8225

0.00000 0.00001 0.00010

0.4444 0.8453 2.4867

0.4515 0.8460 2.4861

1.59 0.09 0.02

165.7521 146.1170

0.00000 0.00001 0.00010

0.5128 0.8112 2.4356

0.5209 0.8120 2.4342

1.56 0.10 0.06

172.5162 156.5328

0.00000 0.00001 0.00010

0.5781 0.7817 2.3776

0.5873 0.7826 2.3762

1.56 0.11 0.06

178.9093 177.9093 176.9093

0.00000 0.00001 0.00001 0.00002

0.6410 0.6812 0.8979 1.0953

0.6511 0.6830 0.8981 1.0950

1.55 0.27 0.03 0.02

232.8917 231.8917 230.8917 229.8917

0.00000 0.00003 0.00006 0.00009 0.00013

1.4621 1.4979 1.5511 1.6508 1.8786

1.4731 1.5008 1.5465 1.6328 1.8662

0.75 0.20 0.29 1.10 0.66

262.9772 261.9772 260.9772 259.9772 258.9772

0.00000 0.00007 0.00015 0.00022 0.00029 0.00036

2.1027 2.1404 2.1871 2.2501 2.3412 2.4998

2.0980 2.1273 2.1684 2.2238 2.3050 2.4533

0.23 0.62 0.86 1.18 1.57 1.90

309.0305 307.0305 305.0305 303.0305

0.00000 0.00068 0.00131 0.00191 0.00250

3.3363 3.4258 3.5443 3.7139 3.9833

3.2185 3.2979 3.4084 3.5658 3.8315

3.66 3.88 3.99 4.15 3.96

338.1919 334.1919 330.1919

0.00000 0.00538 0.00979 0.01359

4.2455 4.4822 4.8493 5.5484

3.8040 4.0520 4.4301 5.2060

11.60 10.62 9.46 6.58

365.8809 361.8809 357.8809

0.00000 0.09842 0.13896 0.16613

4.8246 5.3122 5.7796 6.4454

2.6667 3.5669 4.4373 5.5109

80.92 48.93 30.25 16.96

151.8581

160.4563

167.7774

174.1869

179.9093

233.8917

263.9772

311.0305

342.1919

369.8809

superheated mass fluxes. For LXLe, the proposed method produces good results compared to measured mass fluxes of both saturated and subcooled initial conditions, with the data generally indicating underpredictions approximately between 20% and 55% for qualities close to saturation but decreasing to less than 10% for qualities further away from saturation. Corresponding o-method results are in very close agreement with those of the proposed method in pr0o0.5, but diverge markedly in pr040.5 without the prescribed adjustments to those calculations. The proposed method’s results exceed the graphical data of ASME VIII/1, Fig. 11-2M, for subcooled water ranges by about 50% at 1.2 MPaa to about 4.25% at 21 MPa-a (the ERM vastly exceeds the results of the other techniques in pr0o0.5). For LoLe, the simplified-HNE method produces generally superior results to those of the proposed and the o-method, but

only as long as proper and accurate plim values are obtained from sources apart from the HNE method, of which the proposed method itself is one. However, an inspection of the normalized mass-flux differences C (Eq. (36)) for all L suggests patterns of alignment between the results of the proposed and HNE techniques based on p0, L and x0; study of these comparative patterns could be useful in nozzle-capacity analyses and lead to improvements in predictive methods. It should be noted that various influences presented by the fluid and the overall emergency-relief system design can produce variations in the actual relief flow through nozzles irrespective of the predictive technique used (Ilic et al.). These include the presence of impurities and noncondensibles in the fluid, the general arrangement and heat-transfer characteristics of the system’s equipment, the pressure drop between reservoir and

Table 7 Comparison of proposed, omega, and HNE methods for subcooled water (Sozzi & Sutherland data) Temperature T0 (C)

Initial quality x0

Measured mass flux GcMeas (kg/ s cm2)

Isentropic model mass flux GcIsen (kg/s cm2)

HEM model mass flux GcHEM (kg/ s cm2)

HNE model mass flux GcHNE, PLim ¼ PSat (kg/ s cm2)

(GcIsenGcMeas)/ GcMeas (%)

(GcHEMGcMeas)/ GcMeas (%)

(GcHNEGcMeas)/ GcMeas (%)

C ¼ (GcIsenGcHNE)/ GcMeas

Low or high subc.

0.0 0.0 0.0

6.4810 6.5500 6.6879

6.0808 5.5185 5.1213

276.4945 270.2163 265.4801

0.00050 0.00130 0.00200

6.4937 7.5825 7.5825

2.7465 4.0523 4.9930

2.6920 3.9787 4.9280

4.6425 5.7363 7.0708

57.71 46.56 34.15

58.54 47.53 35.01

28.51 24.35 6.75

0.2920 0.2221 0.2740

L H H

38.1 38.1 38.1 38.1 38.1

6.6189 6.4259 6.7775 6.8533 6.6741

6.2294 5.4453 5.4709 5.3337 4.7855

278.0792 269.3628 269.6630 268.0460 261.2510

0.00050 0.00120 0.00170 0.00200 0.00240

5.1022 5.2609 5.8102 5.8102 5.7125

2.7654 3.9407 4.5449 4.9084 5.4991

2.7099 3.8829 4.4806 4.8406 5.4345

3.9893 4.9355 5.5656 5.9147 6.4110

45.80 25.09 21.78 15.52 3.74

46.89 26.19 22.88 16.69 4.87

21.81 6.18 4.21 1.80 12.23

0.2399 0.1891 0.1757 0.1732 0.1596

L H H H H

63.5 63.5 63.5 63.5 63.5 63.5

5.7088 6.2259 6.3638 6.5086 6.5844 6.7361

4.9610 5.8815 5.5636 5.2986 5.1495 4.9471

263.4888 274.3219 270.7376 267.6269 265.8247 263.3133

0.00080 0.00041 0.00097 0.00150 0.00180 0.00230

3.3201 4.3454 4.6384 4.7849 5.0778 5.5172

3.4543 2.6392 3.5564 4.3819 4.7777 5.3434

3.4118 2.5928 3.5024 4.3214 4.7148 5.2782

4.1917 3.5850 4.3559 5.1488 5.5134 6.0279

4.04 39.26 23.33 8.42 5.91 3.15

2.76 40.33 24.49 9.69 7.15 4.33

26.25 17.50 6.09 7.61 8.58 9.26

0.2221 0.2176 0.1724 0.1603 0.1449 0.1241

H L H H H H

114.3 114.3 114.3 114.3 114.3

6.5362 6.6396 6.7016 6.7706 6.8395

6.3000 5.7775 5.5347 5.6186 5.4708

278.8227 273.1665 270.4031 271.3688 269.6614

0.00030 0.00110 0.00150 0.00150 0.00180

3.8572 4.3454 4.3454 4.6872 4.7604

2.6342 3.6845 4.2947 4.2638 4.6530

2.5921 3.6257 4.2312 4.1994 4.5860

3.2609 4.2008 4.5986 4.5907 4.8597

31.71 15.21 1.17 9.03 2.26

32.80 16.56 2.63 10.41 3.66

15.46 3.33 5.83 2.06 2.08

0.1625 0.1188 0.0699 0.0698 0.0434

L H H H H

190.5 190.5 190.5

6.5844 6.6189 6.8120

5.9513 5.5137 5.5138

275.0893 270.1607 270.1621

0.00080 0.00140 0.00170

3.6585 4.2722 4.7390

3.1542 4.1802 4.5298

3.1161 4.1187 4.4637

3.6436 4.2375 4.4851

13.78 2.15 4.41

14.82 3.59 5.81

0.41 0.81 5.36

0.1338 0.0134 0.0094

H H H

228.6 228.6 228.6

6.6327 6.6741 6.7706

6.3227 6.2887 6.2390

279.0598 278.7042 278.1806

0.00040 0.00050 0.00070

3.3933 3.5642 3.5642

2.7052 2.7735 2.9530

2.6562 2.7178 2.8832

3.1091 3.2263 3.4445

20.28 22.19 17.15

21.72 23.75 19.11

8.38 9.48 3.36

0.1190 0.1270 0.1379

L L L

ARTICLE IN PRESS

Saturation pressure Ps at T0 (MPa-a)

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Stagnation pressure P0 (MPa-a)

Nozzle length (mm)

577

ARTICLE IN PRESS

0.1499 0.1489 0.1682 0.1472 0.1569 0.1537 0.1573 0.1124 0.1449 0.0993 0.1298 0.0385 0.0268 0.0252 -0.0224 0.0219 38.70 12.70 18.04 17.81 8.81 7.29 7.27 3.10 8.58 4.96 7.31 0.65 0.99 2.97 2.92 3.49 149.9 138.5 149.5 149.7 149.3 148.4 150.0 137.6 150.7 135.6 149.2 119.7 119.7 120.4 119.1 120.0 0.5020 0.4050 0.5280 0.5600 0.5620 0.5560 0.5780 0.4700 0.6080 0.4680 0.6140 0.4050 0.5120 0.5590 0.5720 0.6100

0.00001 0.00001 0.00001 0.00002 0.00002 0.00003 0.00003 0.00003 0.00003 0.00004 0.00004 0.00005 0.00009 0.00011 0.00012 0.00013

1.53 1.44 1.59 1.88 1.76 1.76 1.78 1.71 1.95 1.76 2.07 2.05 2.48 2.59 2.66 2.75

0.7101 1.0424 1.0370 1.2714 1.3255 1.3579 1.3699 1.5697 1.5036 1.6683 1.6513 1.9826 2.4393 2.5982 2.6750 2.7862

0.9399 1.2568 1.3047 1.5488 1.6009 1.6277 1.6497 1.7618 1.7868 1.8426 1.9202 2.0615 2.5058 2.6635 2.7345 2.8463

0.7110 1.0451 1.0372 1.2714 1.3215 1.3577 1.3697 1.5695 1.5032 1.6680 1.6505 1.9830 2.4395 2.5982 2.6753 2.7862

53.69 27.59 34.85 32.53 24.50 22.65 23.00 8.14 23.07 4.97 20.29 3.20 1.69 0.45 0.68 1.30

53.63 27.41 34.84 32.53 24.73 22.66 23.01 8.15 23.09 4.99 20.33 3.19 1.68 0.45 0.69 1.30

C ¼ (GcIsenGcHNE)/ GcMeas (GcHNEGcMeas)/ GcMeas (%) (GcHEMGcMeas)/ GcMeas (%) (GcIsenGcMeas)/ GcMeas (%) HEM model mass flux GcHEM (kg/s cm2) HNE model mass flux GcHNE, PAvail ¼ PSat (kg/s cm2) Isentropic model mass flux GcIsen (kg/s cm2) Measured mass flux GcMeas (kg/s cm2) Initial quality x0 Subcooled temperature (C) Pressure (Mpa-a)

Table 8 Comparison of proposed, omega, and HNE methods for subcooled water (Bolle et al. data)

H H H H H H H H H H H H H H H H

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Low or high subc.

578

nozzle, the nozzle’s entrance contour, and even such mechanical characteristics as pipe-metal elasticities that can make systems prone to flow-area distortion. The design engineer might have to take these factors into account in the design of nozzle-based emergency-relief systems. The proposed isentropic enthalpy-difference method for the relief of superheated steam, saturated steam–water mixtures, and subcooled water shows good agreement with other established predictive methods within demonstrated boundaries. With its added advantages of ease of use without reliance on external databases, versatility, generation of complete process data in both tabular and graphical form, high transparency, and stand-alone availability in laptops or personal computers, it can be a valuable resource for engineers engaged in the design of safety relief systems. References Bolle, L., Downar-Zapolski, P., Franco, J., & Seynhaeve, J. M. (1996). Experimental and theoretical analysis of flashing water flow through a safety valve. Journal of Hazardous Materials, 46, 105–116. Darby, R., Meiller, P., & Stockton, J. (2000). Relief sizing for two-phase flow. In 34th annual loss prevention symposium, Session T6004, AIChE Spring Meeting, Atlanta. Darby, R., Meiller, P. R., & Stockton, J. R. (2001). Select the best model for two-phase relief sizing. Chemical Engineering Progress, 97(5), 56–64. Darby, R., Self Jr., F. E., & Edwards, V. H. (2001). Methodology for sizing relief devices for two phase (liquid/gas) flows. In: Process plant safety symposium, AIChE National Meeting, Houston. Fauske, H. K. (1985). Flashing flows or: Some practical guidelines for emergency releases. Plant/Operations Progress, 4(3), 132–134. Fauske, H. K. (1999). Determine two-phase flows during releases. Chemical Engineering Progress, 95(2), 55–58. Fisher, H. G., Forrest, H. S., Grossel, S. S., Huff, J. E., Miller, A. R., Noronha, J. A., Shaw, D. A., & Tilley, B. J. (Eds.). (1992). Emergency relief system design using DIERS technology. New York: The Design Institute for Emergency Relief Systems of the American Institute of Chemical Engineers. Henry, R. E., & Fauske, H. K. (1971). The two-phase critical flow of one-component mixtures in nozzles, orifices, and short tubes. Journal of Heat Transfer, 93(Series C (2)), 179–187. Huff, J. E. (1985). Multiphase flashing flow in pressure relief systems. Plant/ Operations Progress, 4(4), 191–199. Ilic, V., Banerjee, S., & Behling, S. (1986). A qualified database for the critical flow of water Report NP-4556. Palo Alto: Electric Power Research Institute (EPRI). Keenan, J. H., Keyes, F. G., Hill, P. G., & Moore, J. G. (1978). Steam tables: Thermodynamic properties of water including vapor, liquid, and solid phases (SI units). New York: Wiley. Leung, J. C. (1986). A generalized correlation for one-component homogeneous equilibrium flashing choked flow. A.I.Ch.E. Journal, 32(10), 1743–1746. Leung, J. C., & Grolmes, M. A. (1988). A generalized correlation for flashing choked flow of initially subcooled liquid. A.I.Ch.E. Journal, 34(4), 688–691. Leung, J. C., & Nazario, F. N. (1990). Two-phase flashing flow methods and comparisons. J. Loss Prev. Proc. Ind., 3, 253–260. Mandatory Appendix 11, Capacity Conversions for Safety Valves (2004). ASME boiler and pressure vessel code VIII, division 1. New York: The American Society of Mechanical Engineers. Sozzi, G. L., & Sutherland, W. A. (1975). Critical flow of saturated and subcooled water at high pressure. In Report NEDO-13418. San Jose: The General Electric Company. Thompson, L., & Buxton Jr., O.E. (1979). Maximum isentropic flow of dry saturated steam through pressure relief valves. In Third ASME national congress on pressure vessels and piping, San Francisco.