mittoxic industrial chemical (tic) source emissions modeling for pressurized lique!ed gases

8/13/2019 MitToxic industrial chemical (TIC) source emissions modeling for pressurized lique!ed gases

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Author's personal copy

Review

Toxic industrial chemical (TIC) source emissions modeling for pressurizedlique ed gases

Rex Britter a, Jeffrey Weil b, Joseph Leung c, Steven Hanna d , *

a Massachusetts Institute of Technology, Cambridge, MA, USAb University of Colorado, Boulder CO, USAc Leung Inc., Rancho Palos Verdes, CA, USAd Hanna Consultants, 7 Crescent Ave., Kennebunkport, ME 04046, USA

a r t i c l e i n f o

Article history:Received 23 February 2010Received in revised form15 August 2010Accepted 9 September 2010

Keywords:Chlorine releases to atmosphereHazardous chemicalsPressurized lique ed gas releasesSource emission modelsToxic industrial chemicalsTwo-phase jets

a b s t r a c t

The objective of this article is to report current toxic industrial chemical (TIC) source emissions formulasappropriate for use in atmospheric comprehensive risk assessment models so as to represent state-of-the-art knowledge. The focus is on high-priority scenarios, including two-phase releases of pressur-ized lique ed gases such as chlorine from rail cars. The total mass released and the release duration aremajor parameters, as well as the velocity, thermodynamic state, and amount and droplet sizes of imbedded aerosols of the material at the exit of the rupture, which are required as inputs to thesubsequent jet and dispersion modeling. Because of the many possible release scenarios that coulddevelop, a suite of model equations has been described. These allow for gas, two-phase or liquid storageand release through ruptures of various types including sharp-edged and pipe-like ruptures. Modelequations for jet depressurization and phase change due to ashing are available. Consideration of theimportance of vessel response to a rupture is introduced. The breakup of the jet into ne droplets andtheir subsequent suspension and evaporation, or rainout is still a signi cant uncertainty in the overallmodeling process. The recommended models are evaluated with data from various TIC eld experiments,in particular recent experiments with pressurized lique ed gases. It is found that there is typicallya factor of two error in models compared with research-grade observations of mass ow rates. However,biases are present in models estimates of the droplet size distributions resulting from ashing releases.

2010 Elsevier Ltd. All rights reserved.

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. The type of rupture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2. Fluid property information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Release rate and exit thermodynamic conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3.1. Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Recommended approach where the liquid is subcooled referenced to the saturation temperature at both the storage pressure andatmospheric pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.3. Recommended approach where the vapor is superheated referenced to the saturation temperature at both storage and atmospheric pressure 53.4. Recommended approach for the release of material that is initially, or becomes, two-phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6

3.4.1. Release from storage at saturated conditions as a liquid, as a gas or as a two-phase mixture of the two . . . . . . . . . . . . . . . . . . . . . . . . . . . . .63.4.2. The non-equilibrium flow of a saturated liquid through a rupture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .83.4.3. The release of a subcooled liquid through a rupture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .93.4.4. A non-equilibrium flow of a subcooled liquid through a rupture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.4.5. The effect of a rupture in a longer pipe connected to a vessel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.4.6. An exact alternative exact approach for the two-phase case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.5. The discharge coefficient C D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

* Corresponding author. Tel.: 1 207 967 4478.E-mail address: [email protected] (S. Hanna).

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4. Storage vessel response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. On level swell and discharge exit conditions during vessel blowdown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.2.1. Basic relations in level swell analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.2.2. Complete vapor e liquid ( V /L) disengagement (top vent) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4.2.3. Partial V /L disengagement (top vent) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.2.4. Exit conditions for side and bottom vents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135. Flow after exit from the storage vessel and coupling with an air flow and dispersion model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5.1. Jet depressurization to atmospheric pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.2. Flashing of superheated liquid to vapor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.3. Breakup of liquid to form small droplets and rainout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5.3.1. Initial drop size and its distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166. Evaluation of the modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.1. Discharge conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1. Evaluation of the modeling of the mass discharge rate: single-phase release . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .176.1.2. Evaluation of the modeling of the mass discharge rate: two-phase release . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186.1.3. Evaluation of the modeling of the thermodynamic state of discharge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196.1.4. Evaluation of the modeling of the exit velocity and momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

6.2. Evaluation of the modeling of jet breakup, droplet particle size and rainout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206.2.1. RELEASE model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2. RELEASE/CCPS experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3. RELEASE model results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.4. JIP model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.5. JIP model results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7. Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1. Introduction

In this paper, models are addressed that can calculate toxicindustrial chemical (TIC) emission rates and other sourceparameters at the point where the emissions are handed off toatmospheric transport and dispersion models. A few representa-tive papers from this journal describing TIC transport anddispersion models are Anfossi et al. (2010), Hanna et al. (2009),Lee et al. (2007), Mohan et al. (1995) , and Witlox and McFarlane(1994) . These authors usually depend on someone else toprovide the TIC source emission rates and physical and chemicalconditions. In the case of two-phase TIC jets from pressurizedlique ed containers, the transport and dispersion model is usuallylooking for source input information after the velocity of themomentum jet has decreased to a value close to ambientvelocity.

Generally what are required are TIC source emissions modelsthat are of the right quality, i.e., models that are t-for-purpose .

This requirement refers to the incorporated physics and chemistry,to the mathematical coding and to a formal evaluation of theperformance metrics of the model. In the 1996 AIChE Guidelines forVapor Cloud Modeling ( Hanna et al., 1996 ), the available sourceemissions models are reviewed and recommendations made.However, since that time, there have been several advances inunderstanding the scienti c issues concerning these types of releases, and concerns about pressurized lique ed gas releaseshave increased due to a few major railcar accidents and due toterrorism threats. The generic approach for the modeling of thesource term for a TIC release is outlined below.

There are various speci c scenarios of interest and these dependupon the following factors:

The stored material e Each TIC material has its own thermody-

namic, uid mechanical and transport properties, and the degree of detail within various models may require more or less propertyinformation.

The type of storage ( geometry ) e Models may be required thatcan accommodate any shape of storage vessel. However twoparticular shapes are fairly common, that is the cylindrical storagevessel with either vertical or horizontal axis.

The type of storage (thermodynamic state ) e Generally there areonly a limited number of cases and these are:

Material stored as a vapor only; consequently the materialmust be saturated or superheated (with respect to the satu-rated state at the storage pressure); e.g., water vapor/steamstored at 120 C at 1 atm pressure. (See Glossary for the de -nition of terms)

Material stored as a liquid only (though typically stored undera pad gas (e.g., nitrogen) pressure); the material must besaturated or subcooled (with respect to the saturated state atthe storage pressure). This includes the case of material storedas a liquid at ambient temperature and pressure; e.g., waterstored at 20 C and 1 atm pressure. Alternatively it could be

a material stored as a liquid at a pressure that is equal to orabove the saturated vapor pressure for the storage tempera-ture; e.g., water stored at 100 C and 1.2 atm pressure.

Material stored as a vapor and liquid together at saturationcondition, that is at a pressure and temperature that lie on thesaturated vapor line; e.g., water and steam stored together at100 C and 1 atm pressure.

1.1. The type of rupture

There are ve broad categories:

Catastrophic failure

Those that resemble a sharp-edged ori ce; the term

sharp-edged refers to an idealized ori ce type that is often consid-ered as a limiting case.

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Those that resemble many sharp-edged ori ces Those that resemble a short pipe. This last case may be pipe-like just because of the width of the wall of the containingvessel.

Those from a rupture in a long pipe connected to a vessel

Three accidental chlorine releases from rail cars have beenstudied in some detail (e.g., Hanna et al., 2008 ). At the Macdona andGraniteville accidents the ruptures were jagged holes of roughdimension 10 cm by 30 cm. These would be classed as resemblingsharp-edged ori ces unless the storage vessel wall was thicker than0.1 m. The storage vessel wall was estimated to be of the order of 0.1 m for these accidents. For the Festus accident the rupture was atthe end of a long, narrow pipe connected to a storage vessel.

From knowledge of the stored material, the type of storage(geometrical and thermodynamic) and the type of rupture themodel(s) must calculate:

The release rate and other characteristics of the release at therupture

The resulting response of the material within the storage vesselfollowing the rupture

The consequences of the near source depressurization toatmospheric pressure

The partition of the material into vapor, suspended liquidaerosol and liquid rainout close to the release

Our approach requires the selecting, from the various availablemodels, models that will address adequately the processes above.An important issue here is to ensure that the selected models arewell-balanced in the sense that they all have similar accuracy andsimilar speeds and similar complexity or simplicity. These are oftenmatters of judgment and pragmatism and should be guided by theoverall model purpose and also by the current state of develop-ment or development intent of existing operational models.

Our detailed project reports ( Britter et al., 2009; HannaConsultants, 2009 ) cover a comprehensive set of TIC source emis-sions equation recommendations, including evaporation fromliquid pools, evaluations of all equations with eld and laboratorydata, and worked examples from eight categories of TIC scenarios.In this paper we focus on pressurized lique ed gases and considerthe following parts of the modeling process in turn:

Fluid property information Release rate and exit thermodynamic conditions Vessel response Flow after exit from the storage vessel

2. Fluid property information

The TICs are principally of concern as inhalation hazards. Thescale over which health effects occur will depend upon the toxicityof the material, the amount of material likely to be released in anincident, and whether their boiling point is lowenough so that theyrapidly become gases or suspended aerosols upon release. Thereare many toxic chemicals stored as gases and these must also beconsidered; however, the mass stored is likely to be much less thanfor the storage of lique ed gases. Hanna Consultants (2006) indi-cate the comparative health impacts in one particular region thatmay result from releases of various stored and transported TICstaking into account the above features. The top 13 are:

Chlorine, Sulfur Dioxide, Ammonia (anhydrous), Hydrogen

Chloride (anhydrous), Phosgene, Bromine, Acrolein, HydrogenCyanide, Chlorine Tri uoride, Methyl Chloroformate, Oleum (65%SO3), Hydrogen Fluoride, Phosphorus Trichloride

It is clear that common chemicals such as chlorine, ammoniaand sulfur dioxide top this list, re ecting the large amounts of thechemicals being stored and transported. Because many TICs of interest are commonly stored as liquids and will commonlydisperse as gases, the study of TIC releases will also commonly

involve the uid mechanics and thermodynamics of multi-phaseows. The material properties play an important role in such ows.The nature of much of the uid information required is obtained

within the liquid e vapor phase diagram. Various forms of thisdiagram are available of which the most easily interpreted is thatusing pressure and speci c volume (inverse of density) as axes andtemperature as the parameter. However, because pressure andtemperature are not independent in a two-phase region this formof phase diagram is less useful. The preferred version is the tem-perature e entropy form with pressure as the parameter, shown inFig.1, which applies to chlorine. The quantitative data are availablein tabular/electronic form from various standard sources and someof these are listed in the next section.

The lines shown are constant pressure lines in units of Mega-Pascals. The hump shape in the middle is the saturation linethat demarcates the two-phase region. A depressurization, ata commonly-assumed constant entropy, will produce a vertical lineindicating a temperature reduction. The area to the left of the hump is referred to as subcooled liquid while the area to the rightof thehumpis referred toas superheated vapor. These terms areusedhere solely to indicate that the material is a liquid or a vaporrespectively and is not within the two-phase region or on the satu-ration line. These terms are often used when referring to the state of the material within a storage vessel. To the left of the curve thestorage temperature is lower than the saturation temperature atthe same pressure while to the right the temperature is greaterthan the saturation temperature at the same pressure. Here we areusing the terms subcooled and superheated to describe the storageconditions. These terms are also used when the temperature of the

uid is compared with the saturation temperature at ambient(atmospheric) pressure also known as the normal boiling point. Thiswill affect the state of the material as it exits from the storage vessel.If the uid temperature is above the normal boiling point then the

uid is superheated and will change phase upon depressurizationdown to atmospheric pressure. If the uid temperature is less thanthe normal boiling point then the uid is subcooled and will notchange phase upon depressurization down to atmosphericpressure.

Fig. 1. Temperature e entropy diagram for chlorine. Dashed and dotted lines arepressure in units of MegaPascals.

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So a material could be a subcooled liquid relative to saturationconditions at the storage pressure but a superheated liquid relativeto saturation conditions at atmospheric pressure. The terminologyused throughout this paper is to explicitly state whether thereference is to the saturation temperatureat the storage pressureor

at ambient pressure.Becausepressure and temperature are not independent variables(for single-component uids) whentwo phases arepresent, the two-phase region in Fig. 1 collapses to a single curve when the phasediagram is presented on pressure e temperature axes as in Fig. 2.

In this plot the hump of the two-phase region in Fig. 1 hascollapsed to the single line. Note that the red dot in Fig. 2 showsmaterial that is a subcooled liquid relative to the saturation line (atthe same pressure) but is a superheated liquid relative to thesaturation line (at ambient or atmospheric pressure).

Clearly it is important to have access to accurate values of TICproperties. The National Institute of Standards and Technology(NIST) database is currently used in many models for many mate-rials. Alternatively, DIPPR is a widely-used property database.Currently NIST is a sponsor of DIPPR 801 (BYU DIPPR 801 verwebsite). Where the chemical is not found in either NIST or DIPPR,then one has to resort to estimation methods. In this case signi -cant discrepancies may arise depending upon how the estimationsare made. Properties of interest include.Antoine coef cients (forsaturation vapor pressure)

Molecular weightLiquid density (plus variation with temperature) Speci c heats(liquid and vapor)Surface tension (for droplet deformation)Liquid viscosity (for droplet absorption into substrate)

3. Release rate and exit thermodynamic conditions

For most cases with a rupture of a container full of pressurizedlique ed gas, the mass release rate will vary strongly with time(Hanna et al., 1996 ). The optimum input for the dispersion model isthe release rate as a function of time with a resolution of a fewseconds or even smaller for large ruptures. However, manydispersion models cannot use that much detail, and the mass

release rate will, at best, be a set of piecewise constant values, witheach piece averaged over say 1 e 10 min or more. Some models cantake only a constant release rate (although perhaps for a niteduration) or a single total mass released. Thus the total massreleased is also a major parameter. The thermodynamic state of the

material at the exit of the rupture is required as an input to thesubsequent jet and dispersion modeling. Also because the ow maybe choked at the exit the pressure at the exit may be aboveatmospheric pressure. Consequently the jet material will accelerateafter the exit. This requires the calculation of the momentum owrate of the jet including any acceleration due to an elevated pres-sure at the exit plane.

3.1. Background

In order to calculate the mass ow rate from the storagecontainer and the thermodynamic condition of the releasedmaterial at the rupture plane it is common to consider the owasinFig. 3. Within the storage container there exists essentially stagnant

uid and this acceleratestowards the rupture plane. Thepressure atthe rupture plane (or at the vena contracta if one exists) will beatmospheric pressure if the ow is unchoked or will be aboveatmospheric pressure if the ow is choked. The pressure at therupture plane is an important input to the subsequent jet modeling.

We are particularly interested in going from the stagnant(stationary, stagnation) conditions o to those at or just after therupture plane 1 . In all scenarios the pressure will reduce from thestorage pressure to atmospheric pressure. However this may occurdirectly to atmospheric pressure at the rupture plane or it mayoccur with an important intermediary pressure at the ruptureplane that is above atmospheric pressure (a choking condition).In this case there would be a further depressurization down toatmospheric pressure downstream of the rupture plane. The mass

ow rate of a compressible uid through a rupture is often thoughtto depend upon the pressure difference across the rupture. Thisstatement is only partially correct.

For a xed pressure in the storage vessel the mass ow rate doesincrease for smaller and smaller downstream pressures. Howeverthere is a limiting mass ow rate that beyond which the down-stream pressure has no in uence. In this case the mass ow ratedepends on the pressure in the storage vessel and not on thedifference in pressure across the rupture. This condition is called choked ow and is commonly met in single-phase high speed gasdynamics. The maximum exit velocity will be the sonic velocity of the uid (of the order of several 100 s of m s 1) at the thermody-namic conditions at the rupture plane. If the uid is made up of gasand liquid then this can be considered a pseudo compressible uidand will also experience a choking condition. Somewhat

Fig. 2. The saturation curve of pressure against temperature for chlorine; de nition of terms.

Fig. 3. Schematic diagram to illustrate nomenclature. De nition of ow planes o , 1

and 2 .



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surprisingly the maximum exit velocity fora two-phase uid can bemuch smaller than the sonic velocities of either phase individually.Physically this is because the sonic velocity depends on thecompressibility and the inertia (density) of the uid. For the two-phase ashing uid the phase-change process in effect increases

the two-phase mixture compressibility, hence yielding a low sonicvelocity. Maximum exit velocities of the order of only several 10 s of m s 1 are often found.

The types of releases are categorized as coming from containersin which material is stored as a:

1. Liquid in a subcooled state (with a storage temperature that islower than the saturation temperature at the storage pressure).This includes refrigerated lique ed gases (stored at atmo-spheric pressure) and pressurized and partially refrigeratedlique ed gases (often done with anhydrous ammonia). It isimportant to note that if the material is subcooled then, byde nition, there must be an imposed pad gas.

2. Gas in a superheated state (with a storage temperature that ishigher than the saturation temperature at the storagepressure).

3. Either liquid or gas under saturation conditions (at the ends of the horizontal lines in Fig. 1) or liquid and gas together(somewhere along the horizontal lines in Fig.1). The lastcase iscommonly met with both liquid and gas being present withinthe storage vessel. What is being considered speci cally here isthat, during the release, part of the liquid and vapor within thevessel may mix and appear to be like a stored mixture atsaturation conditions.

It is useful to consider the list of TICs mentioned earlier andrefer to them in the three categories above. The 13 TICs on the listare inhalation toxics that are shipped in large quantities. Themost dangerous category of TICs includes those with low boilingpoints (i.e., they are gases at ambient conditions) that aretransported as pressurized liquids. The top three in this groupare chlorine, ammonia (anhydrous), and sulfur dioxide, which allhave boiling points less than 10 C and are shipped in largequantities in rail cars. Other high-priority TICs in this groupinclude chlorine tri uoride, hydrogen chloride anhydrous,hydrogen uoride, and phosgene. As an example chlorine istypically stored under saturation condition with liquid and vaportogether. Consequently the pressure and temperature are notindependent. The pressure in the storage vessel will be thatdetermined by the temperature of the contents of the vessel. If the contents increase in temperature due to heat transfer fromthe storage vessel to the contents then some of the contents willchange phase to a gas and thereby increase the pressure until thetemperature and pressure are back together on the saturationline. Some of these chemicals may be stored as partly pressurizedand partly refrigerated, and some may have a pad gas introducedunder particular situations.

A less dangerous category of TICs on the list includes those thatare liquids at ambient conditions (e.g., acrolein, bromine, hydrogencyanide, and methyl chloroformate), which are most likely to beemitted to the atmosphere primarily by evaporation from a liquidpool. The mass is emitted over a relatively long time period and thedownwind concentrations are relatively low. These four liquidswere selected for the high-priority list because of their relativelylow toxic concentration criteria and the fact that they are trans-ported in relatively large amounts. These are subcooled liquidsunder both storage conditions and ambient conditions.

Finally, two of the TICs on the list are also liquids at storage andambient conditions (e.g., oleum (65% sulfur trioxide) and phos-phorous trichloride)are in the category of fuming liquid , meaning

that they may react with water vapor in the atmosphere, with therelease of heat and conversion to secondary chemicals.

We see that only the rst set of chemicals carries across tocategories 1 and 3 above. The source emission models for releasesof the other chemicals involve liquid pool formation and subse-

quent evaporation.Most importantly, it may well be the case that more than onetype of release may be evident in the one incident and the onerupture. During depressurization of the storage vessel the vesselmay contain regions of vapor only, liquid only and a two-phasevapor e liquid mixture. Depending on the geometric location of therupture different release types will occur. The evolution of theinternal state of the storage vessel during depressurization maymean that a xed geometrical location of the rupture will experi-ence three or more different release types. For example, considera mid-level rupture in a storage vessel that contains material storedunder saturation conditions and predominantly liquid. Initially therupture will be in the liquid space. After rupture the tank contentsmight appear as a liquid phase at the bottom, a vapor phase at thetop and a liquid e vapor mixture between the two. As time proceedsand the tank empties the released material (at the rupture plane)may be all liquid, a two-phase mixture and all vapor.

For the cases labeled 1) and 2) above (and provided the materialdoes not change phase prior to exiting the rupture) and when therelease process is one that always maintains thermodynamicequilibrium, the release problem is relatively straightforward.

3.2. Recommended approach where the liquid is subcooledreferenced to the saturation temperature at both the storage pressure and atmospheric pressure

In this case the liquid in the storage vessel is initially ata temperature below its normal boiling point (that is below the

saturation temperature at ambient pressure so there is no phasechange upon release). It must be stored under a pad gas which is ata pressure higher than ambient pressure. The relevant equation isthe Bernoulli equation

G C D 2 P 0 r f gh P a r f 1=2

(1)

where G mass ow rate per unit rupture area (kg s 1 m 2)r f liquid density in storage tank (kg m 3) P 0 storage pressure(N m 2) P a atmospheric pressure (N m 2) C D discharge coef-

cient (dimensionless); discussed in detail in Section 3.5 g acceleration due to gravity (m s 2) h height of liquid abovethe rupture position (m)

3.3. Recommended approach where the vapor is superheatedreferenced to the saturation temperature at both storageand atmospheric pressure

For the superheated vapor case, the relevant equations are thecompressible gas ow equations and these allow for the importantphenomenon of choked or critical ow. Consider a gas jet comingfrom a rupture in the vapor space of a pressurized liquid storagetank. When the rupture occurs, the velocity, and the mass ow rate,of the exiting uid will depend on the pressure inside and thepressure outside of the container. When these pressures are not toodifferent the ow is unchoked. However the gas can exit therupture only as fast as its sonic velocity and at larger pressuredifferences there is a choked condition at the exit. Choked ow

simply means that the gas is moving through the puncture at itsmaximum possible speed, namely the local speed of sound in thegas.



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The discharge rate equations for an ideal (note that non-idealgas behavior may be expected when the ow approaches thesaturation curve) gas jet release are well known. Whether or notthe ow is choked depends on the ratio of atmospheric pressure totank pressure relative to the critical pressure ratio as de ned

below:

r crit P aP 0 crit

2g 1

gg 1

(2)

where P 0 absolute tank (stagnation) pressure (N m 2)P a absolute ambient atmospheric pressure (N m 2) g gasspeci c heat ratio c p/c v (dimensionless)

For a typical gas, values of g range from 1.1 to 1.5, and r crit valuesrange from 0.5 to 0.59. Thus, the releases of most diatomic gases(g 1.4) will be a choked or critical ow at a critical pressure ratioof 0.528 (i.e., a tank pressure above approximately twice atmo-spheric pressure). Note that the critical pressure ratio r crit will begiven the symbol hc when addressing two-phase ow later.

For an ideal gas exiting through an ori ce under isentropicconditions, the gas mass discharge rate is given by:

_m GAh (3)

G C D ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 P 0v 0 gg 1"P aP 02g P a

P 0

g1g #v uutfor subsonic or unchoked flows

P aP 0

r crit (4)

and

G C D ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP 0v0

g 2

g

1

g1g 1s for choked flows P aP 0

r crit

(5)

where P 0 absolute tank (stagnation) pressure (N m 2),P a absolute ambient atmospheric pressure (N m 2), g gasspeci c heat ratio c p/c v (dimensionless),

_m gas discharge rate(kg s 1), C D discharge coef cient (dimensionless), Ah holeor rupture area (m 2), n0 speci c volume of gas within thevessel

Some care is needed when considering ows that come near tothe saturation curve. The use of perfect gas laws will not beapplicable even if there is no phase change. Furthermore if therelease changes phase (e.g., a superheated vapor cools as itdepressurizes), then it may condense and the condensation willaffect the ow. In addition caution should be shown by the user

about non-ideal gas behavior that will occur at high pressures. Notethat the u -method addressed in Section 3.4.1 can be used in thecase with condensation phenomena.

3.4. Recommended approach for the release of materialthat is initially, or becomes, two-phase

There areseveral approximatemethods available to describe thedepressurization of a material that is initially two phase or thatbecomes two phase during depressurization that do not havedemanding data requirements. Most of these rely on treating thetwo-phase uid as a compressible pseudo- uid whose compress-ibility can be determined based on the bulk properties of the two-phase uid. Several scenarios can be envisaged for which a model

for the source emission rate is required. The two main categoriesare: i) Storage at saturated conditions as a liquid, as a gas or asa mixture of the two, and ii) Storage as a liquid at subcooled

conditions but such that the material would be superheated atatmospheric pressure.

3.4.1. Release from storage at saturated conditions as a liquid,as a gas or as a two-phase mixture of the two

The model recommended here is the u -method ( Leung, 1986,1990, 1996 ). This is a widely-used simple model and has beenextensively tested (e.g., see Richardson et al., 2006 ). The model canbe used forestimating ow rates and exit pressures.The initial statecan be all (saturated) liquid, all (saturated) vapor or a mixture of thetwo but saturated conditions must apply. The u -method was rstproposed for ashing two-phase ow in 1986, and in subsequentyears was extended to non- ashing two-phase ow ( Leung, 1990 ),subcooled liquid ow ( Leung and Grolmes, 1988 ), two-phase pipe

ow ( Leung and Grolmes, 1987; Leung and Ciolek, 1994 ), and thelocal speed of sound in two-phase mixture ( Leung, 1996 ). The u-method is an (isentropic) homogeneous equilibrium model (HEM)for the ow of a saturated two-phase mixture initially at a storagepressure P 0 equal to P v(T 0), the vapor pressure at the storagetemperature T

0.

The u -method is based on three approximations orassumptions:

The liquid e vapor mixture is homogeneous; i.e., there is no slipbetween the phases

The phases are in thermodynamic equilibrium The expansion is isentropic. This last assumption will likely bevalid prior to the uid exiting through the rupture; between 0 and 1 in Fig. 3.

It is applicable to cases where the initial state is two phase or onthe saturation line as a liquid or a vapor. For example it can treat thecase of a saturated vapor; a case in which condensation wouldoccur through the ow passage. As with Section 2.4 where thecompressible gas equations were appropriate, here the two-phasematerial might be considered as a type of compressible gas .Consequently the phenomenon of choking will also be relevant.That is, the ow may be choked (quite common for TIC releases) orunchoked depending upon the material properties and the storageconditions. Because the sonic velocity for a two-phase mixture istypically well below the speed of sound for either the liquid or thevapor phase, smaller exit velocities are expected for the two-phasescenario.

The critical or choked ow rate for various storage conditionscan be found once the pressure P versus speci c volume ( v) rela-tionship is available as tables or formulas. This can be obtainedfrom isentropic ashing calculations at varying pressure, as dis-cussed in the DARPA report ( Hanna Consultants, 2006 ), in theAIChE/DIERS report ( Fisher et al., 1992 ), and by Richardson et al.(2006) . In fact, this is the essence of an exact approach for thetwo-phase case and an example of this is provided at the end of thisSection 3.4.6. However this exact approach requires uid propertydata that may not be commonly available and adequately accuratefor a wide range of chemicals.

The u -method is based on a one-parameter approximation of the P versus v relationship using:

v

v 01 u

P 0P 1 (6)

and

u h a 0 1 2P 0v fg 0

h fg 0 !T 0 c p0 P 0v 0v fg 0h fg 0!

2

(7)



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where the subscript 0 denotes the values in the tank at stag-nation condition, and, a 0 volume fraction of TIC vapor, c p0 heatcapacity of the (liquid) material at T 0 (J kg 1 K 1), h fg0 heat of vaporization at T 0 (J kg 1), P downstream pressure (N m 2),P 0 stagnation (storage) pressure (N m 2), T 0 storage tanktemperature (K), v speci c volume of the material at P (m

3kg

1)vo speci c volume of the material at P o (m 3 kg 1) v fg0 speci c

volume difference between liquid and vapor at T 0 (m 3 kg 1), andvo v fo xo v fgo where xo is the inlet vapor mass fraction or quality

Leung describes the parameter u as consisting of two distinctterms: the rst re ects the compressibility of the two-phasemixture due to the existing vapor/gas volume, and the secondconcerns the compressibility due to phase change upon depres-surization or ashing. The larger the value of u , the more volumeexpansion or compressible behavior the mixture exhibits. As notedin Leung (1996) , ashing ow occurs for u > 1 and non- ashing

ow for 0 < u < 1. u depends only on conditions at stagnation inthe tank. The u -method is presented in the design chart in Leung(1986, 1990) and Fig. 4. The critical mass ux per unit rupturearea, G

c , and the critical pressure, P

c (the pressure at the exit plane

for choked ow), can be found as follows:

1. Find u from its de nition equation above2. Find hc from Fig. 4 or from a solution to the critical pressure

ratio hc

h2c u u 21 hc 22u 2 lnhc 2u 21 hc 0 (8)

3. Find Gc * from Fig. 4 or

a) For choked ow simply use the following formula

G*

c hc = ffiffiffiffiup (9)b) For unchoked ow where h > hc

G*

2u ln hu 11 h0 :5. u

1h

1 1

where h P aP 0

(10)

4. Then P c and Gc are found from

P c hc P 0 (11)

Gc G*

c ffiffiffiffiffiP 0n0s (12)Note that explicit formulas for the critical pressure ratio have

been advanced by Simpson (2003) , and more recently by Moncalcoand Friedel (2010) .

The following gure is assumed to apply to all two-phase uids;an assumption supported by the data presented in Fig. 2 of Leung(1986) ; the gure covers chlorine, ammonia, carbon tetrachloride,water and several other chemicals.

The temperature along a saturation line (as shown in Fig. 2 forchlorine) can be found using the Clapeyron equation ( Perry andGreen, 1997 ). That is

dT =dP v fg T =R fg (13)Thus the temperature T c corresponding to P c can be found.

Additionally since most chemicals should have accurate vaporpressure e temperature relationships or correlations, one could infact be more precise in using them fordetermining the temperatureT c corresponding to P c at the choking location.

The vapor quality (vapor mass fraction), x, at temperature T T c is:

x T h fg

c pf lnT oT x0

h fgoT o

(14)

From the vapor mass fraction the density or the speci c volume

of the two-phase uid can be calculated.

v m 1 xv f xv g v f xv fg (15)If the chemical has reasonable enthalpy data, the ash calcula-

tion can be performed directly without resorting to the aboveapproximate equation.

Note that x, the vapor mass fraction, is related to the vaporvolume fraction, a , by

a xv g

v(16)

In summary, we calculate u and then determine hc and Gc * . These

are then converted to the critical mass ow/unit rupture area Gc and the exit plane pressure P c which are the required unknowns.We can also calculate the temperature at the exit plane and thevapor mass fraction there, and thus the uid density or speci cvolume and nally the exit velocity at the rupture plane.

Note that there is no discharge coef cient in the above analysis.A discharge coef cient of unity could be assumed though see Leung(2004) and Section 3.5 for further guidance.

The above set of equations is generally applicable to releases of material under saturated conditions whether saturated liquid,saturated vapor or saturated two-phase storage.

There are several widely-used simpli ed methods such as theEquilibrium Rate Model (ERM) model by Fauske and Epstein (1988) .Though this approach is now thought to be less appropriate andmore limited than the u -method for the problems at hand the

results for that model are included here as they may be useful forvery simple calculations. The ERM can be derived from the u

method above with the initial vapor fraction ( a 0) set equal to zero.Fig. 4. Design chart for critical ow parameters of a two-phase mixture; that is,initially at saturation conditions.



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GERMz h fgo. n fgo ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT oc pfor dP =dT sat ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT o=c pfor (17)where GERM mass ux for equilibrium rate model (kg s 1 m 2),h fgo heat of vaporization at T o (J kg 1), v fgo difference in speci cvolume between liquid and gas ( v g

e v f ) at T o (m

3

kg1

),T o storage tank temperature (K), c pfo liquid heat capacity at T o(J kg 1 K 1)

provided that x

10, i.e.,low-quality two-phase ow, hc is about 0.9. So the real difference isonly about 10% between the two models. Since ERM has otherlimitations, such as assuming the critical pressure ratio is 1.0 all thetime, the u -method is preferred.

The actual ERM follows from the original Henry and Fauske(1971) non-equilibrium ow model, and contains two parts in itsoriginal form. The part representing the ashing term would begoverning if eq. (18) were satis ed, which leads to the approxi-mation of eq. (17) , the so-called approximate ERM equation.

3.4.2. The non-equilibrium ow of a saturated liquid througha rupture

When the length of the rupture is very short, say like a sharp-edged ori ce, the stored liquid may be able to exit the storagevessel quickly enough so that thermodynamic equilibrium is notmaintained. There is not enough time for the material to changephase until well after the material has exited the vessel.

The non-equilibrium nature of the ow is often thought of interms of the length of the ori ce , in that a sharp-edged ori cewill have a small length and short residence time while the owthrough a pipe-like ori ce will have a long length and a long resi-dence time. The latter ow will appear more like an equilibrium

ow and the former like a non-equilibrium ow. A delimiter of a length of 0.1 m is often quoted.

The use of a dimensional criterion rather than a dimensionlessone is often queried. However experimental data support thedimensional result ( Moody,1975; Fauske,1985 ). The residence timeis of primary importance, but other physical properties such assurface tension and Jakob number could play a part in the bubblegrowth dynamics. Moody argued that the bubble growth time istypically about 1 ms, and that that residence time would translate

approximately to a 0.15 m length. Among the short nozzle data,there are very few datasets that would allow one to examine theeffect of L (or L/D) in detail in the < 0.1 m region; two exceptions areillustrated later in Figs. 7 and 8 .

A model for this type of problem is sometimes called a Homo-geneous Non-Equilibrium (or Frozen) Model (HNEQ). There are alsomore complex homogeneous non-equilibrium models.

For ow lengths less than 0.1 m (the non-equilibrium regime),the ow rate increases strongly with decreasing length. As thelength approaches zero an all-liquid ow occurs and

Gc ;L0 C D 2P s P a=v f 01=2

(19)

where C D discharge coef cient (dimensionless and typically 0.6for an ori ce), P s vapor pressure at the stagnation/storagetemperature T 0 P (T 0) (N m

2), P a ambient pressure (N m2),

n f0 speci c volume of the liquid at saturation condition

This is the Bernoulli equation for an incompressible liquid withpressure drop, ( P s e P a), available for ow. In this case thestorage pressure and the saturation pressure are equal so(P s e P a) (P 0 e P a).

Fig. 5 taken from Fauske (1985) shows the equilibrium and non-

equilibrium results for the mass ux (for saturated conditions).When L is large the mass ow rates are small and the ERM is within20% of the more exact HEM. Much larger release rates are apparentwhen L is small and the ow near the rupture is a non-equilibriumone.

Fauske (1985) proposed an interpolation formula linking theERM model and the non-equilibrium model for the discharge rateof saturated liquid and produced the supporting Fig. 6 below:

A similarinterpolation was developed by Leung between the u

method (as our recommended example of an HEM model) and theori ce model by de ning a non-equilibrium parameter N in termsof the ow length L as:

N Gu =Gc ;L021 L=LE L=LE for L LE (20)

and

G Gu for L > LE (21)

G Gu =N 1=2 for L LE (22)where Gu and Gc,L0 are mass ux per unit rupture area given by theu -method and by the all-liquid ori ce model respectively. Forequilibrium ow to be attained, the ow length has to exceed 0.1 m(a relaxation phenomenon) and the corresponding C D is taken tobe1.0 (i.e., with no vena contracta).

As rst pointed out by Fletcher (1984) and Moody (1975) , theproper scaling in this non-equilibrium region is the pipe length andnot the length-to-diameter ratio ( L/D) of the pipe. A constant

relaxation length ranging from 75 mm to 150 mm has beenproposed by these early researchers. Figs. 7 and 8 illustrate thepresent correlation for the saturated discharge of water and Freon,respectively using a constant equilibrium length of 100 mm. Boththe equilibrium regime and the non-equilibrium region are illus-trated here. For the saturated water case, an inlet pressure P o of 55 bar was used in the prediction (a lower P o would raise theprediction due to the higher liquid density).

When there is full non-equilibrium and L approaches zero thevelocity of the jet exiting the rupture plane will be determined

Fig. 5. Typically ashing discharge data of initially saturated water.



10/26


directly from the Bernoulli equation. There has been some argu-ment about what the pressure at the rupture plane (or the venacontracta) must be. It appears that it must be atmospheric pressure,with the exiting liquid in a non-thermodynamic equilibrium state.The important point is that the jet will not accelerate further due toany pressure excess at or downstream of the rupture plane. Therewill be an expansion, and hence acceleration, due to the subsequent

ashing process. What the exit plane pressure will be in the regionwhere L is less than 0.1 m has not been clari ed.

This would obviously be relevant if the rupture was to be in theform of a short pipe attached to the storage vessel; i.e., thebreakage of pipework attached to the storage vessel.

3.4.3. The release of a subcooled liquid through a ruptureThe u -method, as described earlier, is restricted to materials

that are initially at saturation conditions, be they vapor only, liquidonly or vapor and liquid together. The case of subcooled liquids isnot treatable with that approach. However Leung and Grolmes(1988) and Leung (1995) do provide a modi ed form of the umethod for subcooled liquids. Subcooled liquids may arise due tocooling of the material or to the padding of the stored materialwith another gas. The subcooled liquid release is of particularinterest since it is expected that even a moderate subcooling maycause a much larger mass ow.

The extension of the u -method to subcooled liquid discharge isfacilitated by removing the rst term in equation 7 and de ninga saturated u parameter as:

u s C poT oP s

v fo

v fgo

h fgo!2

(23)

where P s vapor pressure corresponding to inlet temperature T o(N m 2), v fo liquid speci c volume (m 3 kg 1)

Again all the properties are to be evaluated at T o, the inletstagnation temperature. The generalized solutions are divided intolow and high sub-cooling regions, delineated by a transition satu-ration pressure ratio ( P s/P o)t :

hst hP sP o t

2u s1

2u s

(24)

In the low-subcooling region where hs (P s/P o) > hst or P s > hst P o,the uid attains ashing (two-phase) ow prior to the exit plane(throat) location and the dimensionless mass ux is given by

G

ffiffiffiffiffiffiffiffiffiffiffiffiP o=v foq n21 hs2hu shs ln hsh u s 1hs hio

1=2

u s hsh 1 1(25)

and for choking conditions to occur, the following equation yieldsthe critical pressure ratio hc :

u s 1u s 22h

s

h2c 2u s 1hc u shslnhc h

s

32

u shs 1 0 (26)Note that above two equations, Eqs. (25) and (26) reduce exactly

to Eqs. (9) and (10) in Section 3.4.1, respectively, for the saturatedliquid inlet case with hs 1 (i.e., P s P o).

In the high sub-cooling region where hs < hst or P s < hst P o, novapor is formed until the exit plane is reached, and Eq. (25) reducesto a Bernoulli-type formula,

Gc

ffiffiffiffiffiffiffiffiffiffiffiffiP o=v foq ffiffiffiffiffiffiffiffiffiffi21 hsp (27)

or

Gc

ffiffiffiffiffiffiffiffiffiffiffi2r fo

P o P s

q (28)

where r fo 1/v fo, the inlet liquid density at T o. The criticalpressure ratio is given by

Fig. 6. Saturated F-11 mass ux variation with L/D for D 3.2 mm.

Fig. 7. Prediction of saturated water discharge using present correlation (data aspresented in Fletcher, 1984 ).

Fig. 8. Prediction of saturated Freon discharge using present correlation (data aspresented in Fletcher, 1984 ).



11/26


hc P sP o

(29)

or choking simply occurs at the saturation pressure correspondingto the inlet temperature T o. Fig. 9 illustrates the graphical solutionsfor inlet subcooled liquid discharge. The point of consequence hereis that the pressure at the rupture exit plane is the saturation vaporpressure corresponding to the storage temperature and this will bein excess of ambient pressure. Thus the phase change occursoutside the storage vessel.

In Fig. 9 all-liquid discharge will be expected where high sub-cooling is indicated. Where low-sub-cooling is indicated therewill be some phase change within the vessel and a reduced mass

ux discharge. Roughly speaking, if the storage pressure is morethan about 10% larger than the saturation pressure at the storagetemperature an all-liquid release is expected; re ecting themaximum discharge rate possible.

As the stagnation pressure approaches the saturation pressure atthe storage temperature the equation for the strongly sub-cooledmass ux is not consistent with that obtained based on an homo-geneous equilibrium model (HEM) or an (ERM) model for saturatedliquid storage conditions. A pragmatic solution to this is to use theinterpolation proposed by Fauske and Epstein (1988) , where the vector sum is taken of,say, theERM model and strongly sub-cooledmodel above. Fauske and Epstein (1988) present three experimentalresults that, they argue, support the formulas. However it is thoughtthat this may underestimate the ow rate for slightly subcooledliquids andDIERS/AIChE( Fisheret al.,1992 ) recommends themethodgiven in Leung and Grolmes (1988) . But we note again that thesearguments and the interpolation will not produce an exit planepressure. Of considerable importance is that the u-method, whenextended to subcooled materials, provides both the discharge rateand the exit plane pressure, as well as a smooth transition. Of coursefor an all-liquid release that remains all liquid after the exit or one

that is not in equilibrium, then the pressure in the jet will be atmo-sphericat the venacontractaposition for a sharp-edged ori ceand islikely to be atmospheric at the exit plane for a more generic opening.

3.4.4. A non-equilibrium ow of a subcooled liquid througha rupture

A similar non-equilibrium correlation as suggested for thesaturated liquid discharge can be employed for the subcooled liquid

case. Here the equilibrium mass ux is evaluated using the sub-cooled liquid extension to the u method as presented in Section3.4.2 , and is closely approached when L 0.1 m. When Lapproaches zero length the ori ce will appear as a sharp-edged oneand a liquid discharge with a typical C D of 0.6 would be applicable.

3.4.5. The effect of a rupture in a longer pipe connected to a vesselIf the pipework was longer then calculation of the release rate

would need to account for frictional losses in the pipe and thesewould depend upon L/D and the reduction in ow rate due to thephase change occurring within the pipe rather than close to theend. Of course the exit pressures and velocities would be alteredalso. Fauske and Epstein (1988) provide simple estimates of theeffects of friction for modest pipe L/D on the mass ow rate fromthe storage vessel and the exit pressure that may be useful directlyor as an indicator of the magnitude of this effect.

The u -method has also been extended to similar discharges butthrough more extensive pipework andfortwo-phase inlet conditionsincluding saturated liquid and subcooled liquid ( Leung and Grolmes,1987; Leung and Ciolek, 1994 ). For the former the analytical resultscan be illustrated by Fig. 10 where the effect of L/D e shown inhorizontal axis as 4 fL/D where f is the Fanning friction factor e is toreduce the mass ux Gc relative to the equilibrium nozzle case G oc.

As for the subcooled liquid pipe discharge, the bounding equilib-rium ow is given by a simple Bernoulli-type formula ( Leung, 1995 ),

G ffiffiffiffiffiffiffiffiffiffiffiffi2r foP o P s1 4 fL=Ds (30)By bounding it is meant that this will be the maximum

discharge rate; a maximum when the nucleation occurs at the veryend of the pipe and so it is essentially an all-liquid ow throughoutthe pipework.

3.4.6. An exact alternative exact approach for the two-phase caseAn interesting and possibly operationally useful approach has

been provided by Richardson et al. (2006) , who made the veryimportant, though not new, observation that the simpli edapproximate solutions for this problem as provided by Fauske andEpstein (1988) and, more substantially, by Leung (1996) may be anunnecessary approach. Direct and exact calculation may be just asef cient. The simpli ed approaches are of course extremely usefulfor directly determining the relative in uence of the various inputvariables and for determining output trends with the input vari-ables. Simpli ed methods, like the u -method, were meant tocircumvent the hard-to-get thermodynamic properties for indus-trial chemicals. If detailed thermodynamic properties are available

Fig. 10. Critical two-phase pipe discharge based on u -method.Fig. 9. Critical discharge of inlet subcooled liquid based on u -method.



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there be no need to use the u -method , and a classical HEMcalculation such as this can be made.

All that is required for an exact solution is to satisfy threeconditions that connect the ow from the storage tank to therupture/exit plane. The conditions are:

The speci c entropy, s, is the same at both places ( ow isreversible)

The sum of the speci c enthalpy, h, and U 2 is the same atboth places ( ow is adiabatic)

The velocity at the exit plane a 0 is the speed of sound and is

given by ( v P /v r ) 1/2 where the derivative is taken at constantentropy evaluated at the conditions at the exit plane (the owis choked at the exit/rupture plane)

If no solution is found then the ow will be non-choking. In thiscase the condition at the rupture plane should be that thepressure is atmospheric.

The solution relies on an iteration based around the thermo-dynamic properties of the material. These should be accessible formany TICs. A solution procedure is referred to in Richardson et al.(2006) . There is substantial concern regarding the need forconstant entropy ash calculations; the tools for this are available(a robust ash routine or simulation package such as ASPEN) andfurther consideration may be worthwhile depending, in part, onwhat particular scenarios are of interest.

Richardson states that the u -method (a homogeneous equilib-rium model, HEM) is within 1% of exact calculations. The recom-mendation here is that the use of an exact approach should beconsidered further; particularly the availability of the necessarychemical data. Of course, from an engineering standpoint, if the u -method is within 1%, there may be little good reason to useanything more complex!

3.5. The discharge coef cient C D

The dimensionless discharge coef cient, C D, accounts forreductions below the theoretical exit velocity due to frictionallosses arising in the ow near the rupture and the possibility of the

ow area being smaller than the rupture area (a phenomenoncalled the vena contracta). The combination of these two compo-nents means that C D accounts for the reduction in ow rate due toboth the effect of friction and the reduction in the jet area beingsmaller than the rupture area. It depends upon nozzle shape andReynolds number, and graphical relationships of C D for varioustypes of ori ces can be found in handbooks and uid mechanicstextbooks ( e.g ., Perry and Green, 1997 ). In the context of thisdocument, C D is entirely due to the ow eld or vena contractaeffect and is not affected by frictional loss or L/D.

C D is typically 0.6 for incompressible ow of liquids owingthrough circular sharp-edged ori ces. For thin slit ori ces thetheoretical discharge coef cient is p /(p 2) 0.61( Batchelor, 1967 ,p. 360). For sharp-edged ori ces a discharge coef cient of 0.60 isrecommended.

For gas ows through ori ces, the observed discharge coef -cient varies from 0.6 at low pressure drops (the incompressible

ow case) to slightly less than 0.9 at large pressure drops (thecompressible ow case).

Discharge coef cients for a comprehensive range of industriallyrelevant scenarios are available in Wirth (1983) .

Leung (2004) presents results for discharge coef cients forori ce ow and its application to safety relief valves under two-

phase conditions. He also provides theoretical results usingu -method for the discharge coef cient. It varies monotonicallyfrom 0.60 through 0.88 to unity as u varies from near zero (liquid

ow) through unity (near (compressible) gas/vapor ow) to verylarge (saturated liquid inlet). These results have been supportedby the work of Richardson et al. (2006) . Leung (private commu-nication) argues that the saturated liquid case provides a morecompressible mixture due to its ability to ash and greatly

increase its speci c volume, yielding a lesser contraction at thevena contracta location; a plausible argument and consistent withthe u -method.

We recommend:

the use of a value of C D 1 for all scenarios if a truly conser-vative estimate is essential

If this is not the case and a more accurate result is desired thenwe recommend the use of C D 0.60 for all-liquid ows and forall-gas ows where incompressibility can be assumed (at Machnumbers less than 0.3) and the ori ce resembles a sharp-edgedone

For all other cases we recommend the use of C D 1If more detailed information is required then the following may

be considered:

The result from Leung (2004) above Extensive research has been done on discharge coef cients forincompressible, compressible, two-phase uids in various owgeometries ( Jobson, 1955; Bragg, 1960; Chisholm, 1983;Idelchik, 1986; Morris, 1989; Wirth, 1983 ).

The effect of some other geometry can produce dischargecoef cients from below 0.50 for incompressible uids.

Wirth (1983) provides discharge coef cients for many differentshapes of ori ces.

4. Storage vessel response

4.1. Introduction

Following a rupture of a storage vessel there will be a mass owrate from the vessel and this may be gas, liquid or a mixture of gasand liquid depending upon the contents, their disposition and therupture position. The vessel response (called blowdown in thechemical engineering community) will depend upon the funda-mental equations for mass, momentum, enthalpy and entropyincluding heat transfer to or from the vessel contents, transfersbetween different phases within the storage vessel and the ther-modynamic equilibrium or not of the vessel contents. Thesechanges will lead to a change in the phase fractions in the vesseland the temperatures and pressure within the vessel.

The chemical engineering community de nes blowdown as thetime-dependent (transient) discharge from a vent or hole ina pressurized vessel containing lique ed gas, as the tank depres-surizes to ambient conditions. Generally there are several differentscenarios that are considered based on the storage conditions, thethermodynamic properties of the stored material and the type of rupture.

One speci c scenario is of particular interest in the context of this study; the scenario where the material is stored under saturated conditions where both liquid and vapor are presentwithin the storage vessel (and are initially in thermal equilibrium).Under these conditions the storage pressure and the storagetemperature (both assumed uniform) are not independent;knowing the pressure or temperature determines the other (forexample, water boils at 100 C at 101,325 N m 2 pressure). This

scenario is typical of several possible releases of concern. Arelated, though different, scenario is when the storage vessel alsocontains an inert pad gas, such as nitrogen. The nitrogen can be



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used to keep the stored material as solely liquid by ensuring thatthe storage pressure is in excess of the saturated vapor pressurecorresponding to the storage temperature. Often tanks are pres-surized with N 2 in order to unload the tank contents. Evidenceshows that if the blowdown is due to a top opening, then the

pad

gas will be vented out initially and subsequent blowdownwill follow the saturation curve. On the other hand, if the blow-down is from a bottom hole, then the pad gas will exert itsin uence on the discharge rate e a subcooled liquid dischargeregime. Note that the pad gas pressure will continue to diminishdue to the increasing head space as inventory drops. This caneasily be accounted for in a simulation code.

When a tank ruptures or a hole develops in the tank, materialexits the tank and the pressure decreases within the tank. If thehole is well above the vapor/liquid interface then vapor will exitthe tank. If the hole is well below the vapor/liquid interface thenliquid will exit the tank. In both cases the material approachingthe hole, at the hole, and just downstream of the hole willmaintain its state of either vapor or of liquid. The decrease of pressure within the tank will lead to the liquid within the tankchanging phase by nucleation and the liquid boils sometimesviolently. The region of two-phase uid within the tank will benear to the initial position of the uid interface. This region of thetank contents will swell. If the hole is within this two-phaseregion of the tank contents then there will be two-phase uidapproaching, at and immediately downstream of the hole.Experimental data including those of the large-scale DIERSexperiments (500 gal or 2 m 3) demonstrated that the void (orbubble volume) distribution is skew, more void at top than atbottom. However the void pro le would very much depend onthe size of the rupture (opening) and location. A large opening atthe top can induce so much ashing (phase change) inside thetank as a result of rapid depressurization that can cause the levelswell to the top, resulting in a two-phase release.

For an arbitrary hole position and initial uid interface positionvarious time-dependent ows will evolve. For example:

A hole at the top of a tank, well above the vapor/liquid inter-face, will initially discharge vapor. As the pressure in the tankbegins to decrease, some of the liquid will vaporize (andeventually exit the tank) and slow the pressure decrease. Thismay be a substantial fraction of the tank mass. The vaporizingphase change within the tank cools the tank contents by autorefrigeration, and eventually the vapor pressure falls to atmo-spheric pressure and discharge ceases. More commonly therewill be a continued release after pressure has come down toequilibrate with the outside ambient pressure because of theheat transfer to the tank contents from the tank walls and fromoutside the tank. Note that heat transfer from the tank wallsmay in uence the blowdown process but heat transfer fromoutside will be too slow to play an important role in theblowdown phase.

With a hole at the bottomof a tank, well below the vapor/liquidinterface then the discharge will be all liquid. It will remain anall-liquid release until the two-phase region within the tankdescends to the position of the hole.

If the hole is initially near the liquid/vapor interface there willbe a two-phase release closely following the rupture and thiswill remain so until the two-phase region within the tankdrops below the hole position.

The overall picture is best considered for a hole initially well

below the vapor/liquid interface. The sequence of events (at theplane of the rupture) would be an initial all-liquid release, thena two-phase release and then a single-phase release. The pressure

would remain relatively constant but slightly decreasing until allthe liquid had vaporized. The mass ow rates would be largest forthe all-liquid release, less forthe two-phase release and smallest forthe single-phase vapor release. The exit velocities (and possibly themomentum uxes) would be in the reverse order. The all-liquid

release produces the smallest velocities and the all vapor releasesproduces the largest velocities.The most prevalent release scenarios involve pressurized liq-

ue ed gas of TICs such as chlorine, anhydrous ammonia, and sulfurdioxide, which would lead to two-phase jet releases. Two-phase

ow stops when the inventory depletes suf ciently for the frothylevel to drop below the bottom of the puncture. The blowdownstops when the pressure in the tank reaches atmospheric.

An example of a particularly simple blowdown model (from Joseph C. Leung) follows. This is included in this paper as indicativeof what might be developed.

4.2. On level swell and discharge exit conditions during vesselblowdown

The purpose of this section is to summarize the methodologyand underlying equations describing the level swell and thedischarge exit conditions during a vessel blowdown. The discussionis necessarily brief and should provide a good-enough rstapproximation to the rather complex in-vessel hydrodynamicbehavior during blowdown.

In this treatment the model assumptions are:

(1) Neglect transient level swell associated with pressure under-shoot or delayed bubble nucleation during the initialblowdown.

(2) Thermodynamic equilibrium between vapor and liquid phases.(3) One-dimensional model allowing vertical void distribution.(4) Constant cross-sectional area vessel.(5) Uniform vapor generation from blowdown (depressurization).(6) Churn-turbulent ow regime with C o (void distribution

parameter) of 1.0 in terms of the drift- ux model.(7) Uniform pressure and temperature in-vessel

4.2.1. Basic relations in level swell analysisThe level swell inside a vessel is a buoyancy-driven ow

phenomena where the rising of swarms of vapor bubbles throughthe liquid bulk results in the level swell. This phenomena is bestdescribed by the so-called drift- ux model ( Wallis,1969; Zuber andFindlay, 1965 ). Applications of the drift- ux model can be found inblowdown analysis of nuclear reactor components and chemicalequipments. The DIERS work utilized the drift- ux model exten-sively in describing the hydrodynamics of both chemical runawayreactions and storage vessels during pressure relief conditions(DIERS report, 1983; Fisher et al., 1992 ). It suf ces here to justsummarize the key results. During a vessel blowdown, the super-

cial vapor velocity at the top j g N and the average void fraction inthe liquid pool a are related via:

j g NU N

2a1 a

(31)

where U N is the individual bubble rise velocity (m sec 1) given byWallis (1969) .

U N 1:53"s g r f r g r 2 f #1=4

(32)



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Here s is the liquid surface tension (N m 1), r f is the liquiddensity (kg m 3) and r g is the vapor density (kg m 3). In most cases,r g


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processes. Very close to the rupture (within a few meters) theashing and the depressurization of the jet down to atmospheric

pressure occur but without any entrainment of ambient air.Subsequently the jet is assumed to be at atmospheric pressureand entrainment of ambient air takes place. Across these two

arti cial processes the various uxes are required to be contin-uous. Of great importance is that within the near rupture regionthe liquid that has not ashed will be forming small droplets dueto both mechanical shear and the thermodynamic ashingprocesses. Not all of these effects are included in all widely-usedmodels, and sometimes one or both parts of the jet are includedin a source model. Often the second part of the jet is included inthe atmospheric dispersion model. Consequently the transitionfrom the source model to the atmospheric dispersion modeloccurs at different locations for different models. Many publicly-available models ignore some of the jet effects or only roughlyparameterize them. For example, the Lagrangian particle modelfor dense gas dispersion described by Anfossi et al. (2010) doesnot simulate the initial dense jet but needs the jet precalculatedby another model for input to their model. However, theHGSYSTEM model described by Witlox and McFarlane (1994) hasa module for calculating certain types of jets, but not all possi-bilities. The CFD model used by Hanna et al. (2009) for simulatingthe transport and dispersion of chlorine releases in an urban areais looking for jet inputs from an independent source emissionmodel in the version in that paper. However, the CFD modeldevelopers now have an internal model for two-phase jetsincluded in their transport and dispersion model for TICs.

When the concentrations are needed very near the sourcelocation (say, within 100 m) it is necessary to account for more of the source details than if the concentrations are needed at fartherdistances (say beyond a few hundred meters). For example, by thetime the plume reaches a distance of 1 km, the aerosol generationand evaporation processes are much less important, as long as therainout (and reduction of plume mass ux) can be approximated.Sensitivity studies can assist in decided which details arenecessary.

The material exiting through the rupture plane will eventuallyform a conventional momentum jet at atmospheric pressure. Therequirement is to model the ow between the rupture plane andthe momentum jet at atmospheric pressure. This is commonlyattempted by nding an equivalent source condition for themomentum jet that is consistent with the conditions at the ruptureplane. If the ow is unchoked at the rupture plane then the pres-sure is atmospheric and there is no further depressurization. If the

ow is choked at the rupture plane then there will be an acceler-ation of the released material as it depressurizes. Additionally thedepressurization will allow further liquid to change phase toa vapor and cool the jet. This process is commonly called ashing .Accompanying this process will be a breakup of the liquid part of a jet into smaller droplets as a result of mechanical shear instabilityand the thermodynamic ashing process. Finally, there will also bemixing between the momentum jet and the surroundingatmosphere.

All these effects must be subsumed into an alternative initialcondition for the momentum jet at atmospheric pressure. This isoften done by assuming that there is a depressurization to atmo-spheric pressure and there is no mixing between the momentum jet and the surrounding atmosphere until after the depressuriza-tion has been accounted for.

In this section we are considering the processes between planes1 and 2 in Fig. 3, but not allowing any mixing of the surrounding

atmosphere into the jet.There are three distinct though related phenomena here, as

summarized below.

5.1. Jet depressurization to atmospheric pressure

The rst of these is the acceleration of the jet downstream of therupture as the uid depressurizes to atmospheric pressure.Considering a ow from a hole (subscripted 1 ) to the end of

expansion (subscripted

2

), the following approaches are mostlyused in the literature:Mass and Momentum Balance

_mu 1 P 1 P 2 A1 _mu 2 (47)

u2 u1 P 1 P 2 A1_m u1

P 1 P 2G1

(48)

where _m mass ow rate (kg s 1), G1 mass ux (kg m 2 s 1),u velocity (m s 1), P pressure (Pa), A ow area (m 2)

Enthalpy Balance

H 1 12

u21 H 2 12

u22 (49)

where H is enthalpy (expressed as c pT for ideal gases).It should be noted that the development above is commonly

used; however, one aspect of the analysis is open to question. Theargument presented is a one-dimensional one though the ow issketched to, and is observed to, diverge in the downstream direc-tion, particularly close to the source. Actually little detail is knownabout this ashing/mixing/depressurizing ow. Our methodology,which breaks the ow into a depressurization and accelerationregion and a following mixing region is a gross simpli cation. Onbalance the level of analysis used here is thought to be appropriatefor the problem at hand until further experimental information isavailable to support or negate it.

5.2. Flashing of superheated liquid to vapor

The second phenomenon is the phase change from liquid tovapor immediately downstream of the rupture plane. The massfraction that initially changes phase by ashing is determined bydirect thermodynamic calculations. The ash fraction arising froman all-liquid release is x g C plT 0 T b=h fg (50)where C pl is the liquid speci c heat, T 0 and T b are the temperaturesin the storage vessel prior to ashing and the boiling point of thematerial at atmospheric pressure respectively, h fg is the phase-change enthalpy at atmospheric pressure. This formulation may befound in Fauske and Epstein (1988) and may be traced back to theapplication of the steady ow energy equation while ignoring thekinetic energy terms. Their inclusion leads to

x g C plT 0 T b=h fg 1=2 U 21 U 22 .h fg (51)where U 1 and U 2 are the jet velocities at the rupture plane and afterany depressurization to atmospheric pressure has occurred. U 2 willbe equal to or larger than U 1 and thus the neglect of the velocityterms will lead to an overestimation of the ash fraction. The actualusage of this equation is in uenced by whether or not the U 2

term is incorporated within the subsequent jet model or not.The equivalent result when there are two phases present at the

exit plane is:

x g x g 1 C plT l T b=h fg 1=2 U 21 U 22 .h fg (52)where T 1 is the temperature at the exit plane.



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5.3. Breakup of liquid to form small droplets and rainout

The third phenomenon is the breakup of the remaining liquidinto small droplets with a particular probability density functionfor their sizes. The larger droplets will fall out to make a pool on

the ground while the smaller droplets remain suspended and willevaporate in the plume. The characteristic droplet diameter is oftento be determined from correlations. From this the rainout fractionmay be estimated though this should be in uenced by the uiddynamics of both the jet ow from the rupture and the ambientwind conditions.

An important aspect of this section and the next, in the contextof real scenarios, is that the release and jet may have many orien-tations and may hit and be in uenced by various obstructions.Additionally they may not be directed into regions of clear air butmay be submerged within large clouds formedby material releasedearlier and remaining near to the material source. This is verypossible when the local wind speed is small.

In the case of a ashing liquid jet, such as a release of chlorineor ammonia or HF from a pressurized container, the chemical will

ash partially to vapor when released to atmospheric pressure asdescribed above. For ammonia, about 80% of the released massremains as liquid. Thus, the quality , or mass fraction that isvapor, is 100% 80% 20%. The liquid component in the aerosol isoften parameterized to consist of spherical droplets with sizesdistributed around some mean diameter. The droplets areassumed to be surrounded by a mixture of air and evaporatedvapor. Some large liquid drops may rainout (i.e., may fall) imme-diately onto the ground, some drops may move under gravity anddrag force while vaporizing and may rainout later, and some mayremain in the plume as an aerosol. However, the fraction of liquidthat rains out versus the fraction that remains suspended in theplume as an aerosol is not well known. This fraction is importantbecause the greatest hazard range is likely to occur when all therelease remains suspended as is the maximum possible sourceemission rate for the subsequent dispersion modeling. Addition-ally the evaporation of the suspended aerosol produces anincreased negative buoyancy and this can affect the downwindplume greatly. The liquid that rains out is likely to form a boilingpool that will continuously evaporate and add vapor mass into thedispersing jet. The evaporation may continue long after the jetrelease ceases. This process is controlled by the heat transfer to theliquid pool.

The breakup of the liquid into droplets occurs by two mecha-nisms; a mechanical breakup dominant for releases that are sub-cooled relative to ambient conditions and a thermodynamicbreakup dominant for liquid or two-phase releases that aresuperheated relative to the ambient conditions though both effectsmay also occur together.

The modeling of these processes is complex and the results aretypically in the form of correlations. It is sometimes forgotten thatthe essential information required is the amount of releasedmaterial that rains out. Additionally knowledge of where therainout occurs may be required for input to the pool evaporationmodeling but pragmatically all that may be required is knowledgeof where the mean particle diameter rains out.

Byde nition whatever does not rainout must remain within the jet/plume and be subsequently be evaporated within the jet/plume.Most jet/plume models can accommodate this evaporation processeither in bulk due to air entrainment or by considering evaporationof individual droplets. In the latter case knowledge of the meandroplet diameter of the liquid in the jet/plume or the actual particle

size distribution is required. Whether such detailed knowledge of the evaporation process is required in operational models could beargued.

However the approach commonly taken is to determine a meanparticle diameter, assume a particle diameter distribution and thentrack the fate of particles within the jet/plume downstream of therupture; with some of the particles raining out and some evapo-rating. Of course this all hinges on whether an appropriate two-

phase jet model is available and whether this is correctly linked tothe (possibly multi-phase) plume model.The mean diameter and/or the size distribution of the liquid

drops have to be input to the models for droplet motion andvaporization. In some models, thedropletsize is calculated based onan assumption of an aerodynamic or mechanical breakup mecha-nism (e.g.,the shattering of initial drops into smaller droplets due toaerodynamic forces). The stable droplet size is determined by theratio of the disruptive hydrodynamic forces to the stabilizingsurfacetension force. The ratio is the Weber number, We :

We r g u2rel d p

s (53)

where r g

surrounding gas density (kg m 3), urel

relative

velocity between the jet and gas (ambient air) (m s 1), dp dropletdiameter (m), s surface tension of the liquid (N m 1).It is generally accepted that there is a critical Weber number,

mittoxic industrial chemical (tic) source emissions modeling for pressurized lique!ed gases

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