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Journal of Loss Prevention in the Process Industries 11 (1998) 391–395

Critical mass flow rate in accordance with the omega-method ofDIERS and the Homogeneous Equilibrium Model

T. Lenzinga, L. Friedel a,*, M. Alhusein b,1

a Department of Fluid Mechanics, Technische Universita¨t Hamburg, Eisendorfer Strasse 38, 21073 Hamburg, Germanyb Mu’tah University, Al Karak, Jordan

Abstract

A correlation frequently used in practice for the design of the relief cross-section next to the Homogeneous Equilibrium Modelis the so-calledv-method. For the determination of thev-parameter a definition was originally reported by J.C. Leung in 1986. Anew expression was proposed by the same author in 1995. Predictions of the critical mass flow rate using both the new and theold v-parameter formulations as well as the Homogeneous Equilibrium Model are compared for some typical substances. Resultsdemonstrate that the deviations as a rule are acceptable for practical use, if the proposed range of application and recommendedproperty data calculation are respected. 1998 Elsevier Science Ltd. All rights reserved.

Keywords:v-method; Homogeneous Equilibrium Model; Critical mass flow; Two-phase flow

Nomenclature

cp Specific heat at constant pressureG* Mass flow rateh Specific enthalpyn Exponent of the state changep Pressures Specific entropyT Temperaturen Specific volumex* Flow mass vapour/gas qualityx0 Stagnation mass vapour/gas ratek Isentropic exponentSubscriptsV/G VapourL LiquidG Gashom Homogeneouscrit Fluid dynamic critical condition0 Stagnation condition

* Corresponding author. Tel.:+ 49-40-77183052; Fax:+ 49-40-77182573.

1 Alexander von Humboldt Fellow.

0950–4230/98/$—see front matter 1998 Elsevier Science Ltd. All rights reserved.PII: S0950-4230 (98)00022-9

1. Introduction

Proper calculation of the critical mass flow rate playsa significant role in the fluid dynamic design for two-phase flow duty, e.g. of safety relief systems. Overestim-ation may result in a serious accident by undersizing thesafety valve that would not be able to reduce the pressurein the protected installation. On the other hand, undere-stimation of the critical mass flow rate may lead to over-load in the outlet installation.

The classical Homogeneous Equilibrium Model(HEM) was recently chosen by the DIERS for emerg-ency relief sizing design (Fisher, 1991; Huff, 1985;Leung, 1992). Although this is considered by experts tobe a conservative method with respect to the calculationof the required relief area, evaluation of the HEMinvolves a lengthy (iterative) procedure and requiresdetailed thermodynamic property tabulation (Leung &Nazario, 1990). A generalised correlation for one-component homogeneous equilibrium flashing chokedflow was published by J.C. Leung in 1986 (Leung, 1986)for simplifying this procedure. It has come to be knownas the ‘omega’ method since anv-parameter first pro-posed by Epstein et al. (1983), comprising dimensionlessphysical property groups, was introduced to characterisea wide range of fluid conditions. As only stagnation

392 T. Lenzing et al. /Journal of Loss Prevention in the Process Industries 11 (1998) 391–395

properties are required, this method can be used in mostengineering calculations, especially those where exten-sive property data are not readily available. In 1995, aslightly modified form of thev-parameter was proposedby Leung (1995), which has the attribute of bringing outkey parameters in connection with the compressible nat-ure of a two-phase flow system.

The definition of thev-parameter was originally basedon the assumption of an isothermal state change of thetwo-phase mixture (Leung, 1986). With this, deviationsappear with respect to the critical mass flow rate asopposed to the normally isentropic gas or vapour statechange, as a limiting case of the two-phase flow, of upto 20%. In the modified correlation this limit value ismore adequately included.

So, the question arises of how far this correctionaffects the critical mass flow rate predictions in the com-plete range of the two-phase flow. To answer this ques-tion, the predictions in accordance with both the old andthe newv-parameter formulations, as well as the HEM,were compared for some typical substances and differentoperational conditions. The most essential calculationequations are presented.

2. Critical mass flow rate models

The critical mass flow rate formulations according toboth the HEM and thev-method are presented onlybriefly. However, these equations and the way they arearrived at can be found elsewhere (Leung, 1986, 1995;VDI-Warmeatlas, 1994). In the HEM the ideal case ofa frictionless, adiabatic flow, and the immediate delayfree establishment of saturation values for density andtemperature during the (fast) relief are assumed. Further-more, the two phases flow with the same average velo-city. The mass flow rate is defined as

G* = √2(h(p0) − h(p))/nhom(p)

with

h(p) = x*hV/G(p) + (1 − x*)hL(p)

and

nhom(p) = (x*nV/G(p) + (1 − x*)nL(p)

Hereh is the specific enthalpy,s is the specific entropy,nL, nV/G andnhom are the specific volumes of liquid, vap-our or gas and the homogeneous mixture, respectively,p0 is the stagnation pressure andx* is the mass flowvapour/gas quality. For an isentropic state change thequality is

x* =x0(sV/G(p0) − sL(p0)) + sL(p0)) − sL(pcrit)

sV/G(pcrit) − sL(pcrit)

the stagnation mass qualityx0 is determined as:

x0 =MV/G

MV/G + ML

The critical mass flow rate is obtained by numericallysearching for the maximum value as a function of the(static) nozzle exit pressure: starting at the stagnationpressure the exit pressure is lowered in discrete stepsuntil the critical flow rate is obtained.

As an explicit approximate solution for the HEM thev-method contains a polytropic instead of an isentropicstate change of the two-phase mixture. The critical massflow rate results from the quotient of sound velocitya2ph

and specific volume of the homogeneous mixturenhom

at the so-called fluid dynamic critical pressure

G*crit = a2ph/nhom =

√ − 2p0/nhom(p0)·(v ln(pcrit/p0) + (1 − v)(pcrit/p0 − 1))/

(v(p0/pcrit − 1) + 1)

where the critical pressure can be calculated from

(pcrit/p0)2 + (v2 − 2v)(1 − pcrit/p0)2 + 2v2 ln(pcrit/p0)

+ 2v2(1 − pcrit/p0) = 0

To solve iteratively for the determination of the fluiddynamic critical pressure, the equation can be replacedthrough for thev range from 1 to 100

pcrit = p0(0.55 + 0.217 ln(v) − 0.046(ln(v))2

+ 0.004(ln(v))3)

Thev-parameter characterises the initial compressibilityof the mixture and the increase due to the phase tran-sition. It contains the mean void fraction in the stag-nation state as well as the Jakob number (Epstein et al.,1983) and has the following definition for the oldexpression:

v1986 = x0nV(p0)/nhom(p0) + cpLT(p0)p0/nhom(p0)

SnV(p0) − nL(p0)hV(p0) − hL(p0)

D2

and for the modified correlation is

v1995 = x0nV(p0)/nhom(p0)·S1 −2p0(nV(p0) − nL(p0))

hV(p0) − hL(p0)D

+ cpLT(p0)p0/nhom(p0)SnV(p0) − nL(p0)hV(p0) − hL(p0)

D2

393T. Lenzing et al. /Journal of Loss Prevention in the Process Industries 11 (1998) 391–395

HereT denotes the temperature andcpL is the liquid spe-cific heat.

The expressions can be reduced for two-phase mix-tures with a negligibly small evaporation to

v1986 = x0nG(p0)/nhom(p0)

v1995 = x0/k·nG(p0)/nhom(p0)

For the calculation of the critical mass flow rates at athermodynamic critical pressure or temperature ratio ofmore than 0.5 and 0.9, respectively, the liquid specificheat, which is not usually known in this range suf-ficiently exactly, is replaced by the termDhL/DTup0

(Leung, 1986), in which J.C. Leung (personal communi-cation, 1996) recommends a temperature difference of2 K.

An advantage of thev-method, besides the possibilityof the explicit calculation of the critical mass flow rate,is the low number of required fluid properties. Only atthe set pressure of the safety device are the temperatureand pressure as well as the specific volumes andenthalpies needed. However, this simplification has anadverse effect on the calculation of the specific volume(Simpson, 1995). For example, for an air/water mixtureat a pressure of 50 bar (absolute) and temperature of25°C, the specific volume differs by around 27% and inthe case of a butane gas/liquid mixture at a pressure of30 bar (absolute) the difference is about 14% of the cor-rect fluid property. These inaccuracies lead to consider-able deviations in the calculated values of the criticalmass flow rate compared with the values calculated bythe HEM. The determination of the liquid enthalpy tem-perature gradient also causes difficulties in practice,because the fluid properties of the sub-cooled liquid areneeded, which are only known for a few substances, asa rule.

In the following, the predictions of the critical massflow rates according to the HEM and both correlationsof the v-method are compared for selected two-phasemixtures and operating states.

3. Predictions of the critical mass flow rates

The calculations were made at first for a steam/watermixture at three thermodynamic critical conditions ofpressure and temperature. In Fig. 1 the critical mass flowrate is plotted against the stagnation mass quality at arelative pressure ratio of 0.09. All curves show a similarcourse, starting from the largest mass flow rate at liquidflow, decreasing strongly with increasing stagnationmass quality and approaching asymptotically the lowervalue at steam flow, which is, however, only reached bythe HEM. The latter represents the upper limitationabout the complete range of the two-phase flow, whereas

Fig. 1. Critical mass flow rate according to J.C. Leung’s original andmodified model and the homogeneous equilibrium model as a functionof stagnation quality for water at a thermodynamic critical pressureratio of 0.09.

with the old v-parameter the lower boundary is given.However, deviations between the predictions of the cal-culation methods should, for technical reasons, be negli-gible, especially as the preference for the model’s selec-tion is based on a convention.

At a pressure of 100 bar (absolute), which equals athermodynamic critical pressure ratio of 0.45, the termDhL/DTup0 is used instead of the specific heat. This isbecause the relative temperature ratio of 0.9 as a limitis about to be exceeded. This is also contrary to the rec-ommendation that the critical mass flow rate is appliedby using the specific heat (Leung, 1986). Here, this stilldoes not lead to deviations and the trends of the predic-tions are almost identical as shown in Fig. 2. The differ-ences between the two mass flow rates predicted by thev-method are based on the definition of the parameterin both old and new correlations. As opposed to the criti-cal mass flow rates calculated with the HEM the devi-ations are approximately 6% and 3%, respectively. Once


394 T. Lenzing et al. /Journal of Loss Prevention in the Process Industries 11 (1998) 391–395

more, with the HEM one can get the highest resultswhereas the lowest are obtained by the oldv-method.

Also, as demonstrated in Fig. 3, this order continuesnear the thermodynamic critical condition at a pressureratio of 0.9 (a pressure of water of 200 bar absolute).Again, the HEM represents the upper limit. With the oldand newv-parameter, using three values of temperaturedifference (1, 2 and 5 K) clearly lower results areobtained. The deviations are directly proportional to thesize of the temperature difference. The lower this is, thesmaller the discrepancies between the calculated criticalmass flow rates in accordance with thev-method andthe HEM. The lowest results for the critical mass flowrate are calculated using the liquid specific heat (contraryto the recommendation given in Leung, 1986). They areup to 50% lower than those in accordance with the HEM.The deviations which result from the new expression ofthe polytropic exponent are once more against this byonly about 6%.

On the basis of the fluid properties of the refrigerantR12 the fundamental course of the curves is the sameas before (Figs. 4 and 5). However, at a thermodynamiccritical pressure ratio of 0.9 the differences increasebetween the calculations with increasing stagnation massquality, in which for the case of a vapour flow the pre-dictions in accordance with thev-method are up to 22%too small (Fig. 6). The HEM represents an upper limitonce more. The deviations between the predictions ofthe critical mass flow rates, using the specific heat, arelower here as in the case of steam/water mixture. Never-theless, in principle, Leung’s instruction to use theenthalpy gradient with respect to temperature, as analternative the termDhL/DTup0 is sound.

Calculations for a chlorine vapour/liquid mixture at athermodynamic critical pressure ratio of 0.9 confirm theobservations for water and the refrigerant R12 in termsof the trend of the curves; however, the relative differ-


Fig. 4. Critical mass flow rate according to J.C. Leung’s original andmodified model and the homogeneous equilibrium model as a functionof stagnation quality for refrigerant R12 at a thermodynamic criticalpressure ratio of 0.09.



395T. Lenzing et al. /Journal of Loss Prevention in the Process Industries 11 (1998) 391–395

Fig. 7. Critical mass flow rate according to J.C. Leung’s original andmodified model and the homogeneous equilibrium model as a functionof stagnation quality for chlorine at a thermodynamic critical pressureratio of 0.9.

Fig. 8. Critical mass flow rate according to J.C. Leung’s original andmodified model as a function of stagnation quality and pressures foran air/water mixture.

ences are lower (Fig. 7). In general, for single-compo-nent two-phase flows, a better adjustment of the predic-tions is carried out with the modifiedv-parameter thanone in accordance with the HEM.

Comparing the predictions for the case of an air/watermixture it can be recognized that unlike the previousresults in the range of small to nearly medium meanstagnation mass qualities, the results obtained lie closerto those of the HEM when the oldv-parameter methodis used compared with the new definition (Fig. 8). Thedifferences between the curves corresponding to the oldand the new expressions become larger with increasingpressure; however, they are negligible for technical pur-

poses. However, it is surprising, that the predictions inaccordance with thev-method lie insignificantly overthose obtained with the HEM.

4. Conclusions

In general, it is valid to say that the use of either theold or the newv-parameter correlation could result indeviations between each other in the calculation of criti-cal mass flow rates. These should be negligible in prac-tice, as a rule, if faults of up to 10% are tolerable.

For the fluids considered, the contrary use of the spe-cific heat near the thermodynamic critical state shows,in contrast to the predictions in accordance with thehomogeneous equilibrium model, deviations of up to50% with increasing pressure. With correct use of thetemperature gradient on the liquid enthalpy curve at con-stant pressure the size of the differences calculated forthe critical mass flow rates depends on the temperaturedifference. The value recommended by J.C. Leung of2 K leads in individual cases to deviations of up to 25%as opposed to the predictions of the homogeneous equi-librium model.

Leung (1996) Personal communication not cited intext.

References

Epstein, M., Henry, R. E., Midvidy, W., & Pauls, R. (1983). One-dimensional modeling of two-phase jet expansion and impinge-ment. Second International Topical Meeting on Thermal-Hydraulics Nuclear Reactors, St. Barbara.

Fisher, H. G. (1991). An overview of emergency relief system designpractice.Plant/Operations Progress, 10(1), 1.

Huff, J. E. (1985). Multiphase flashing flow in pressure relief systems.Plant/Operations Progress, 4, 191.

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

Leung, J. C. (1992). Size safety relief valves for flashing liquids.Chem.Eng. Progress, 88(2), 70.

Leung, J. C. (1995). The Omega method for discharge rate evaluation.Runaway Reactions and Pressure Relief Design. Int. Symp., Boston.

Leung, J. C. (1996). Personal communication, September 11th.Leung, J. C., & Nazario, F. N. (1990). Two-phase flashing flow

methods and comparisons.J. Loss Prevention Process Industries,3(2), 253.

Simpson, L. L. (1995). Navigating the two-phase maze.RunawayReactions and Pressure Relief Design. Int. Symp., Boston.

VDI-Warmeatlas (1994). Berechnungsbla¨tter fur den Warmeubergang,Abschn. Lgcl/2: Kritische Massenstromdichte, Du¨sseldorf.

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