CHAPTER 27

MODELING ATMOSPHERIC DISPERSION OF HEAVIER-THAN-AIR CLOUDS CONTAINING AEROSOL Jaakko Kukkonen and Juha Nikmo Finnish Meteorological Institute, Air Quality Research, Helsinki, Finland

27.1

INTRODUCTION Since the early 1970s, numerous mathematical models have been presented for the dispersion of heavier-than-air clouds, and considerable progress has been made both theoretically and experimentally. Reviews of these developments have been written, e.g., by Blackmore et al. (1982), Wheatley and Webber (1984), Hanna and Drivas (1987), Koopman et al. (1989), Bricard and Friedel (1998), and Britter (1998). Clearly, practical modeling applications also require modeling of release rates, source terms, and effects for various conceivable accident scenarios, as shown in Fig. 27.1. Modeling of heavier-than-air clouds involve several processes: e.g., gravity spreading, interaction with the surrounding atmosphere and the ground, cloud transport, the influence

Source term

Effects Dispersion

Releases

e

FIGURE 27.1 Schematic presentation showing modeling of releases, source terms, atmospheric dispersion and effects for various conceivable accident scenarios. (Source: David Webber, Integral Science and Software, U.K.)

27.1

27.2

CHAPTER TWENTY-SEVEN

of terrain and obstacles, deposition of material to the ground, and thermodynamic processes within the cloud. These processes are shown in Fig. 27.2. Several of these processes have been addressed in previous chapters of this Handbook. This article focuses on the influence of aerosol content of the cloud. Releases of liquefied toxic or flammable gases often take place in aerosol form, consisting of vapor and liquid droplets of the released species, together with entrained humid air. This has been demonstrated in several laboratory and field-scale experiments (e.g., Koopman et al., 1986, Moodie and Ewan, 1990; Nolan et al., 1990; Schmidli et al., 1990; Nielsen et al., 1997). Aerosol phenomena may have a significant influence on the temperature and density evolution of the source term and on the subsequent heavy gas dispersion. In particular, the deposition of substance liquid droplets may, under certain conditions, cause a substantial decrease of concentration. However, two-phase dynamics are commonly treated in a very simple way (or even ignored by some modeling systems) by both fairly simple integral models and more complex three-dimensional fluid models. Practically all of these models rely on the homogeneous equilibrium assumptions, implying that the liquid is uniformly distributed in the cloud and that the liquid and the gas are at a uniform temperature and in thermodynamical equilibrium. This chapter presents the mathematical structure of a more general nonequilibrium model AERCLOUD (Kukkonen et al., 1989; Vesala and Kukkonen, 1992). The homogeneous equilibrium model is the limiting case of the aerosol model presented, when the cloud is well mixed and the droplet-gas equilibration processes are rapid compared to the rate at which other processes occur. We then proceed to review results on the applicability of the homogeneous equilibrium assumptions (Kukkonen et al., 1993, 1994; Nikmo et al., 1994). We have compared thermodynamic model predictions with those of the heavier-than-air cloud dispersion model DRIFT (Webber et al., 1992). These investigations have shown that the simpler model does indeed provide a good description for many release situations and guidance is given on where the homogeneous equilibrium model is not likely to be adequate. Mathematical models for droplet vaporization and condensation have also been developed by Hewitt and Pattison (1992), with the aim of incorporating the aerosol model into a heavy gas dispersion model. Woodward and Papadourakis (1991) have presented a two-phase jet model, including a description of droplet vaporization. Woodward et al. (1995) have devel-

Influence of terrain and obstacles Atmospheric processes Entrainment of air Transport of the cloud _ u

Volatile droplets + gas

e

Gravity spreading

Deposition to the ground

Heat flux from the ground

FIGURE 27.2 Processes important for the atmospheric dispersion of heavier-than-air clouds.

MODELING ATMOSPHERIC DISPERSION

27.3

oped a model to simulate the dispersion and trajectories of a high-momentum aerosol jet, including a droplet evaporation model. Muralidhar et al. (1995) describes a thermodynamical nonequilibrium two-phase jet model for predicting the HF releases. Witlox and Holt (1999) have presented a dispersion model including a nonequilibrium thermodynamic model for droplet vaporization. Most of the heavier-than-air cloud field experiments have focused on the dispersion of gaseous clouds. However, some experiments have been conducted with two-phase effects, e.g., the Desert Tortoise, Eagle, Goldfish series (Koopman et al., 1989) and FLADIS field experiments (Nielsen et al., 1997). In particular, the Desert Tortoise series was designed to study the dispersion of two-phase ammonia clouds. It was reported that two-phase effects had a dominating influence on the temperature and spreading behavior of the cloud and the extent of the heavy cloud dispersion regime (Koopman et al., 1986). Recently, available field data sets have been reviewed within the SMEDIS (Scientific Model Evaluation of Dense Gas Dispersion Models) project (Daish et al., 1998; Carissimo et al., 1999). Nolan et al. (1990), Moodie and Ewan (1990), and Johnson (1991) have studied twophase source terms on the laboratory scale. Johnson (1991) has presented results on the deposition of contaminant liquid versus liquid superheat for six substances. Nolan et al. (1990) and Moodie and Ewan (1990) have also presented measurements on droplet characteristics, including size and number concentration.

27.2 27.2.1

THEORY Statement of the Problem

Consider a system consisting of binary (two-component) droplets together with the surrounding gas (such as air) and the vapors of species forming the droplet. In the following we refer to the gaseous phases of the two condensing or evaporating species simply as ‘‘the vapors’’ and to the mixture of air and the two vapors as ‘‘the surrounding gas,’’ or ‘‘the gas.’’ The total mass of humid air within the system is time dependent, while the total mass of contaminant is assumed to be constant, i.e., the deposition of droplets and the contaminant vapor are neglected. We refer to the system of droplets and gas as ‘‘the cloud.’’ The droplets are assumed to contain no impurities, to be spherical, and to be falling freely. They are also assumed to be well mixed; the validity of this assumption has been studied in detail by Vesala (1993). The gas is assumed to behave as an ideal gas. The total number of droplets is constant, i.e., the coagulation of droplets is neglected. The total pressure is assumed to be equal to atmospheric pressure at all times. Finally, the droplet population is assumed to be monodisperse and uniformly distributed in the gas. In the following, we derive equations for the rate of change of quantities related to (1) the droplet population (the total mass of the droplets, the droplet composition, and the droplet temperature) and (2) the gas (the gas temperature, the droplet number concentration, and the gas composition, i.e., the partial vapor pressures in the gas). For a more detailed model description, the reader is referred to Nikmo et al. (1994). Physically, the composition and temperature of the gas around the droplets are constantly changing due to the condensation and evaporation processes and to entrainment. For instance, the evaporation of contaminant tends to cool the gas and increase the concentration of contaminant vapor in the gas. The changes in the composition and temperature of the gas in turn have an influence on the behavior of the droplets. 27.2.2

Change of Quantities Related to the Droplet Population

The Change of the Total Mass of a Droplet. The droplet evaporation or condensation rate is described by means of the mass and heat fluxes from the droplet into the surrounding gas. In the following, the two condensing or evaporating species are denoted with the subscripts

27.4

CHAPTER TWENTY-SEVEN

1 and 2. In a steady state, the mass fluxes are constant according to the equation of continuity, and the quasisteady change of the droplet mass md is by definition equal to the sum of the mass fluxes of species 1 and 2 from the droplet surface into the gas: dmd ⫽ I1 ⫹ I2 dt

(27.1)

The positive radial direction has been chosen to be into the droplet (i.e., condensation corresponds to a positive mass flux). Generally, the mass fluxes depend on the droplet radius, the total gas pressure, the droplet and gas temperatures, the mole fractions of the vapors just above the droplet surface and far from it, and the diffusivities of vapors in the gas mixture. The Change of the Droplet Composition. In order to determine the mass fluxes, the composition of the droplet must be known. A direct consequence of the definition of the mass fluxes (Ii ⫽ dmi / dt, i ⫽ 1,2) is that the change of the mass fraction Xi of species i in the droplet can be written as:

冋

册

dXi d mi I ⫺ Xi(I1 ⫹ I2) ⫽ ⫽ i dt dt m1 ⫹ m2 md

(27.2)

where mi is the mass of species i within the droplet. The Change of the Droplet Temperature. In order to determine the mass fluxes, the droplet temperature must also be known. An equation for the quasisteady droplet temperature can be derived using energy conservation, yielding: dTa L1I1 ⫹ L2I2 ⫹ 4␲ak(Tg ⫺ Ta) ⫽ dt md cd where Li a k Tg cd

⫽ ⫽ ⫽ ⫽ ⫽

the the the the the

(27.3)

heat of vaporization for species i droplet radius thermal conductivity of the gas temperature of the gas average specific heat capacity of the droplet

Kulmala and Vesala (1991) have discussed alternative methods for modeling the droplet temperature, also taking into account the temperature dependencies of the vapor phase enthalpies. 27.2.3

Change of Quantities Related to the Gas

Equations (27.1) to (27.3) form a complete set of equations for describing the evolution of a single droplet in an infinite volume of gas with no entrainment. In the following, the gas temperature and the partial vapor pressures in the gas are allowed to change due to the mutual influence of evaporating or growing droplets and the entraining humid air. It is assumed that the entrained air will be distributed sufficiently rapidly and uniformly into the cloud. The Change of the Gas Temperature. The temperature of the gas changes due to the thermal conduction between the droplets and the gas and due to the entrainment of air, i.e., dTg ⫽ dt

冉

N L1I1 ⫹ L2I2 ⫺ md cd

冊

dTa ⫹ (Tamb ⫺ Tg)cambGent dt cgmg

(27.4)

MODELING ATMOSPHERIC DISPERSION

where N Tamb camb Gent cg mg

⫽ ⫽ ⫽ ⫽ ⫽ ⫽

27.5

the number of droplets the temperature of the ambient gas the average specific heat capacity of the ambient gas the mass flux of air into the gas cloud the average specific heat capacity of the gas is the mass of gas

The Change of the Droplet Number Concentration. The droplet number concentration is defined as the number of droplets in a unit volume of the cloud. The droplet number concentration changes due to (1) the change in the cloud volume caused by the temperature change of the gas, (2) the change in the number of gas molecules caused by evaporation and condensation, and (3) the change in the cloud volume caused by entrainment:

冉

冊

dCN C dT C N I1 I C G ⫽⫺ N g⫹ N ⫹ 2 ⫺ N ent dt Tg dt n M1 M2 nMamb

(27.5)

where n ⫽ the number of moles in the gas Mi ⫽ the molar mass of species i Mamb ⫽ the average molar mass of the ambient air The Change of the Partial Vapor Pressures. The partial vapor pressure changes due to evaporation and condensation processes and entrainment, i.e.,

冉

冊

⫺ p1)Gent dp1 N I1 I (p ⫽ (p1 ⫺ p) ⫹ 2 p2 ⫹ amb,1 , dt n M1 M2 Mambn

(27.6)

where pi ⫽ the partial vapor pressure of species i p ⫽ the total pressure pamb,1 ⫽ the partial vapor pressure of species 1 in ambient air The equivalent equation for species 2 is found by interchanging subscripts 1 and 2 in Eq. (27.6).

27.2.4

Modeling Mass and Heat Transfer from Droplets to Surrounding Gas

Here we do not consider in detail the modeling of the mass and heat transfer processes from the droplet surface into the surrounding gas. The evaporation or condensation of droplets has been addressed in detail by Kukkonen et al. (1989), Kulmala and Vesala (1991), Vesala and Kukkonen (1992), and Vesala and Kulmala (1993); see also Kulmala et al. (1993). Vesala (1991) has discussed the validation of various mass flux and droplet temperature equations against laboratory-scale experimental data.

27.3 27.3.1

NUMERICAL RESULTS Flash Fractions

The flash fraction is a useful concept in interpreting results on two-phase flow. Flashing can be defined as an isentropic evaporation process due to depressurization, in which the final pressure is equal to the atmospheric pressure. The flash-off vapor fraction (flash fraction) is the corresponding quality in the final state of the process. The term partial flashing refers to the same process when the final pressure is higher than the atmospheric pressure.

27.6

CHAPTER TWENTY-SEVEN

During flashing, the total entropy of the fluid is conserved: xv,0 Sv,0 ⫹ (1 ⫺ xv,0)Sl,0 ⫽ xv, ƒ Sv,f ⫹ (1 ⫺ xv, ƒ )Sl,ƒ , where

x S Subscripts 0 and ƒ Subscripts v and l

⫽ ⫽ ⫽ ⫽

the the the the

(27.7)

mass fraction specific entropy initial and final states of the process vapor and liquid phases

We assume that the fluid is in a saturated state initially and finally and that the fluid is initially pure liquid (xv,0 ⫽ 0). Then the flash fraction (xv,ƒ) can be computed numerically from Eq. (27.7). The final temperature of the fluid is the boiling point temperature of the species. Figure 27.3 shows the flash fractions for some commonly occurring toxic and flammable substances as a function of temperature of storage. The curves for propane, chlorine, ammonia, sulfur dioxide, and hydrogen fluoride were computed numerically from Eq. (27.7) and the curves for ethylene and butane have been taken from Marshall (1987). The flash fraction increases with increasing storage temperature and vanishes at the boiling point of species. The flash fractions for the considered substances, except for ethylene and propane, are less than 0.20 for commonly occurring ambient temperatures. The significance of flash fractions is that they give the maximum possible vapor fraction formed in an isentropic process for discharges into the atmosphere. This statement could be derived mathematically starting from Eq. (27.7) and equations for the temperature dependency of the vapor and liquid entropies in saturated state. For instance, in choked isentropic pipe flow the vapor fraction of the fluid at the aperture is always smaller than the flash fraction (because the exit pressure is higher than the atmospheric pressure). Evaporation and Deposition of Ammonia–Water Droplets

We have written a computer program, AERCLOUD (for ‘‘aerosol cloud’’) for solving Eqs. (27.1) to (27.6), and the corresponding mass and heat transfer equations between the liquid

Et hy le ne

0.4

0.3

Flashing fraction

27.3.2

e an op Pr

0.2

rine Chlo onia Amm ide diox fur Sul

0.1

ne ta Bu

en Hydrog fluoride

0 -40

-30

-20

-10

0

10

20

30

Temperature (°C) FIGURE 27.3 Flash fraction versus temperature of storage for some toxic and flammable substances. (Source: Kukkonen, 1990)

MODELING ATMOSPHERIC DISPERSION

27.7

and gaseous components (Kukkonen et al., 1989; Vesala et al., 1989; Kukkonen 1990; Vesala, 1991; Vesala and Kukkonen, 1992; Nikmo et al., 1994). Consider a droplet that consists initially of pure ammonia. Then ammonia evaporates and water vapor condenses into the droplet surface. The evaporation rate of ammonia depends on its vapor pressure at the droplet surface, which is a function of the droplet temperature, the ammonia mole fraction in the droplet, and the activity coefficient. The condensation of water vapor has two simultaneous effects on the evaporation of ammonia, a thermal effect and a dilution effect. In the thermal effect, latent heat is released by the condensing water vapor, which tends to raise the droplet temperature and therefore enhance the evaporation of ammonia. In the dilution effect, the concentration of ammonia in the droplet decreases with increasing water vapor condensation. For a large dilution, the mole fraction of ammonia and the activity coefficient are small, reducing the evaporation rate of ammonia. Figure 27.4 shows the radius of an ammonia droplet as a function of time for different relative humidities. Initially, the thermal effect of water vapor condensation is dominant and the rate of evaporation of ammonia is increased with increasing relative humidity. Later, however, the rate of evaporation of ammonia is slowed down due to the dilution effect. Figures 27.5a and 27.5b show the drying and settling times of water and ammonia droplets versus the droplet radius. The curves have been shown for three combinations of gas temperatures and vapor pressures and for three initial heights. The simultaneous evaporation and settling of droplets have been taken into account in the computation of the curves. Each settling time curve corresponds to one drying time curve, and each intersection point of these curves corresponds to the maximum radius rM of a totally evaporating droplet. The vapor pressures in Figs. 27.5a and 27.5b correspond to saturation ratios 0.0012 and 0.058 at a gas temperature of 20⬚C and to a saturation ratio of 0.12 at a gas temperature of 0⬚C, for both species. The radii rM range from approximately 40 to 110 ␮m for water droplets for the selected ambient conditions and initial heights. The corresponding radii for ammonia droplets range from 90 to 380 ␮m. The larger radii for ammonia droplets are mainly due to the shorter drying times. Evidently, the radius rM increases with increasing release height because the gravitational settling time increases with height. For a given release height, rM decreases with decreasing gas temperature and increasing vapor partial pressure because the drying

100

NH3 T∞ = 20°C pv1∞ = 0 atm

Radius (µm)

80

60

RH = 100% RH = 50%

40

RH = 0%

20

0 0

0.5

1

1.5

2

2.5

3

Time (s) FIGURE 27.4 The influence of relative humidity on the evaporation of an ammonia droplet. The initial droplet radius is 100 ␮m, the gas temperature is 20⬚C, and the vapor pressure of ammonia in the gas is negligible. (Source: Vesala and Kukkonen, 1992)

CHAPTER TWENTY-SEVEN 100

T∞ = 20°C, pv∞ = 2.7·10-5 atm T∞ = 20°C, pv∞ = 1.4·10-3 atm

g yin Dr

tim

T∞ = 0°C, pv∞ = 7.0·10-4 atm

e

Se ttlin gt

H2O

ime

Time (s)

10

10 m 1

3m

1m 0.1 1000

100

Droplet radius (µm)

(a) 100

T∞ = 20°C, pv∞ = 0.01 atm

NH3

T∞ = 20°C, pv∞ = 0.5 atm T∞ = 0°C, pv∞ = 0.5 atm

10

Time (s)

27.8

Dr

g yin

tim

Se ttlin gt

e

im e

e

10 m

1 3m

1m 0.1 100

1000

D roplet radius (µm )

(b) FIGURE 27.5 Drying time and gravitational settling time of water and ammonia droplets versus droplet radius. The curves are shown for three combinations of gas temperature and vapor pressure, and the initial height of droplets ranges from 1 m to 10 m. (Source: Kukkonen et al., 1989)

MODELING ATMOSPHERIC DISPERSION

27.9

time is longer in these conditions. The curves in Figs. 27.5a and 27.5b have been calculated assuming a negligible droplet concentration. The effect of the simultaneous evaporation of a population of droplets is to increase the drying times, resulting in smaller rM. The droplet size regimes in which deposition or total evaporation is likely to take place in various conditions can be estimated from Fig. 27.5a. For instance, for a 100-␮m droplet released from heights of from 0 to 3 m in the given conditions, the settling time is less than about 5 seconds and the drying time is about 20 seconds. The deposited mass fraction is therefore expected to be fairly large (the model computations give Xd ⫽ 0.82). For a 300␮m droplet, the settling time is much smaller than the drying time, and the deposition fraction is therefore expected to be very close to unity (the model computations give Xd ⫽ 0.99). 27.3.3

Evolution of a Two-Phase Ammonia Cloud

We have illustrated the model predictions by evaluating two-phase ammonia clouds released in dry and moist air. The numerical test cases are identical to those in Kukkonen et al. (1993), which presents a comparison of the model AERCLOUD and the thermodynamical submodel of the heavy cloud dispersion program DRIFT (Webber et al., 1992). DRIFT embodies the homogeneous equilibrium model, while AERCLOUD allows also for thermodynamic nonequilibrium effects. Both models will cope with ammonia interactions with moist air as well as with the simpler dry air problem. We emphasize at the outset that our objective is a comparison of concepts—homogeneous equilibrium versus aerosol dynamics. It is done in the context of two models but is not intended to apply only to those models. By keeping everything in the two models the same, apart from the aerosol model, we shall focus sharply on the difference in predictions arising solely from the nature of the aerosol model. We shall use predictions on the mass of entrained air from the code DRIFT. An aerosol cloud is modeled with the same initial content using the code AERCLOUD, importing air into it at the time-dependent rate given by DRIFT. In this way we need not couple the model AERCLOUD explicitly to a heavy gas dispersion model. It is important to distinguish between vaporization and condensation, on the one hand, and deposition phenomena, on the other. The prime objective here is to examine the vaporization and condensation processes, and so we shall ignore deposition in the following, however large the droplets. The gravitational settling velocities of 10, 100, and 1,000 ␮m ammonia droplets in dry air are about 0.01, 0.5, and 5 m / s, respectively. The mathematical model makes assumptions about homogeneity of the cloud; we are assuming homogeneity, or at least an approximately self-similar global inhomogeneity. However, equilibrium is regarded as the most questionable part of the homogeneous equilibrium assumption; in the instantaneously released cloud flow considered here, there is evidence (see, e.g., Brighton, 1985, for a discussion of the Thorney Island Trial data) that the strong frontal gravity current vortices mix the cloud reasonably well, at least in the early stages, when aerosol effects are likely to be most important. The convective fluxes of mass and heat substantially increase the purely diffusive mass transfer and the purely conductive heat transfer for large droplets. The expressions for the convective fluxes are not valid for very high droplet number concentrations. We therefore believe it prudent to make estimates using two model options: including and excluding convective mass and heat transfer. Selection of Cases. We have made predictions for instantaneous releases of a pure ammonia cloud into both dry and humid air. A summary of the initial and ambient conditions for the cases chosen is shown in Table 27.1. The ambient temperature was taken to be 15⬚C, the Pasquill class was D, the average wind velocity at a height of 10 m was 2 m / s, and the roughness length was 6 mm. The total mass of contaminant was taken to be 10,000 kg, which is roughly two or three times the mass of the initial gas cloud in the Thorney Island field experiments (McQuaid

27.10

CHAPTER TWENTY-SEVEN

TABLE 27.1 A Summary of the Cases Selected for the

Numerical Results

Case

Ambient air relative humidity (%)

Solution in liquid phase

Contaminant liquid fraction (%)

01 02 03 11 12 13 21 22 23

0.00 0.00 0.00 99.99 99.99 99.99 99.99 99.99 99.99

– – – Ideal Ideal Ideal Interactions Interactions Interactions

85 60 30 85 60 30 85 60 30

and Roebuck, 1985). The flashing has been assumed to have taken place, and therefore the pressure is atmospheric and the initial contaminant temperature is equal to the boiling point of ammonia (⫺33⬚C). The initial contaminant liquid fraction by mass was assumed to be 30, 60, or 85%. The value of 85% corresponds approximately to the largest possible liquid fraction after flashing has taken place, and the values of 30 and 60% correspond to cases where part of the released ammonia has been stored in vapor form. In order to illustrate the effects of atmospheric moisture, we have included the case of dry air and that of 99.99% relative humidity. Two options were used for the interactions of ammonia and water in liquid phase: assuming an ideal solution and allowing for the actual interactions. Ammonia and water behave attractively in liquid phase, and the activity coefficients are therefore smaller than unity; respectively, the partial mixing enthalpies are negative (Vesala and Kukkonen, 1992; Wheatley, 1987). For an ideal solution, the activity coefficients are equal to unity and the partial mixing enthalpies vanish. Numerical Results. We have evaluated case 21 in more detail. The droplet size regime considered ranges from 100 ␮m to 1,000 ␮m. The curves including and excluding the convective mass and heat transfer have been marked in the figures with ‘‘ventilation’’ and ‘‘no ventilation’’, respectively. The droplets consist initially of pure ammonia; ammonia then vaporizes and water vapor condenses onto the droplet surface. Figures 27.6a, b, and c show the liquid, vapor, and total concentration of ammonia versus time for case 21. The curves are shown for three values of the initial droplet radius, including and excluding droplet ventilation. The total concentration of contaminant is almost identical for these three figures because the numerical tests were designed so that the entrainment of air versus time is the same by definition. These figures show that the ammonia liquid is almost completely vaporized during 200 seconds in all three cases. The influence of droplet ventilation decreases with decreasing initial droplet radius, and for a droplet radius of 100 ␮m (or smaller), the influence of ventilation on the contaminant vaporization is negligible. These figures also show that the influence of the initial droplet size on the vaporization rate of ammonia is fairly small, unless the droplet radius is on the order of a few hundred micrometers or larger. One reason for this phenomenon is that the evaporation / condensation process within a cloud is self-controlling, with negative feedback. For instance, consider the vaporization of fairly small droplets. The vaporization of contaminant is more efficient (compared to large droplets), which causes a lower cloud temperature and a larger contaminant

MODELING ATMOSPHERIC DISPERSION

Concentration (kmol/m)

1 Case 21 100 µm Ventilation

0.1

Total Liquid Vapor

0.01 0.001 0.0001

0

40

80 120 Time (s)

160

200

(a)

Concentration (kmol/m)

1 Case 21 300 µm

Total Liquid Vapor

0.1 ventilation

0.01 no ventilation

0.001

ventilation

0.0001 0

40

80 120 Time (s)

160

200

(b)

Concentration (kmol/m)

1 Case 21 1000 µm

Total Liquid Vapor

0.1 ventilation

0.01 ve

0.001 0.0001 0

40

nti

lat

ion

80 120 Time (s)

no ventilation no v entil atio n

160

200

(c) FIGURE 27.6 Contaminant molar concentration versus time for case 21 for three values of the initial droplet radius (100, 300, and 1000 ␮m). The curves are shown for liquid, vapor, and total concentrations. For an initial droplet radius of 100 ␮m, the curves including and excluding droplet ventilation are indistinguishable.

27.11

27.12

CHAPTER TWENTY-SEVEN

vapor pressure in the cloud. This in turn reduces the rate of vaporization. For large droplets, the changes in the gas properties due to vaporization are smaller, which tends to allow a correspondingly more efficient vaporization.

27.4 27.4.1

HOMOGENEOUS APPROXIMATION Design of the Test

Our strategy for testing the homogeneous equilibrium model will be as follows. First we shall look at some predictions of the code DRIFT for instantaneous releases of ammonia into moist air. Following this, we shall model a gas cloud with the same initial content using the code AERCLOUD, importing air into it at the time-dependent rate given by DRIFT. In this way we can study the differences of model predictions for the thermodynamic behavior, excluding possible differences in modeling, such as entrainment, spreading, or transport processes. We also need not couple the aerosol model explicitly to a heavy gas dispersion model. The HE model is the limiting case of the more complex aerosol behavior when the cloud is well mixed and the droplet-gas equilibration processes are rapid compared to the rate at which other processes occur. An important pretest condition, therefore, is that the two approaches be shown to be in agreement in the limit where the heat and mass transfer coefficients of the aerosol model are set to be sufficiently large to make these processes very rapid, i.e., the values predicted by AERCLOUD at the HE limit are expected to duplicate the respective results computed with DRIFT. This is also a very useful verification of the accuracy of the derivation and coding of each of the models. Both models are fairly complex, and the model equations are solved using different numerical techniques. The models also use different, independently validated parametrizations for the physicochemical properties of substances, in particular for the properties of the mixtures of ammonia and water in liquid phase. 27.4.2

Predictions of the Model AERCLOUD

We are considering here only monodisperse aerosol populations. The initial droplet size is required by the aerosol model as input information; we have chosen the radii of 1,000 ␮m, 100 ␮m, and 10 ␮m. The droplet radius values in the laboratory-scale measurements in this context range typically from a few to a few hundred micrometers (e.g., Nolan et al., 1990; Moodie and Ewan, 1990; Schmidli et al., 1990). The droplets are assumed to consist initially of pure ammonia, and they are at the assumed source term temperature (⫺33⬚C). We have selected as base cases those of dry air and of humid air with ammonia water interactions, both with the liquid fraction of q ⫽ 0.85. These were evaluated in more detail using the model AERCLOUD. Figures 27.7a to d show the computed numbers of moles of ammonia (vapor and liquid phases). The headlines of figures show also the selected initial droplet sizes (1,000 ␮m and 100 ␮m). The curves marked with ‘‘HE-limit’’ show the model predictions in the homogeneous equilibrium limit. First, we compare the HE limit predictions of the model AERCLOUD (in Figs. 27.7a to d) with the corresponding results computed with the model DRIFT. The maximum differences for the numbers of moles of contaminant vapor and liquid, and for the temperature, are smaller than 4%; and for water vapor and liquid water the maximum differences are smaller than 9%. The differences of various modeling options can be seen clearly in Figs. 27.7a to d. The rates of mass and heat transfer are slowest for the nonequilibrium computation, where droplet ventilation has been excluded (‘‘no ventilation’’). Including droplet ventilation in the none-

MODELING ATMOSPHERIC DISPERSION

500

po r

q = 0.85 Dry air NH3, 1000 µm

va

Number of moles (kmol)

600

r po va

400

or vap

300 liquid

200 no ventilation

100

liq

ventilation HE-limit

0

0

40

ui

liq d

uid

80 120 Time (s)

160

200

(a)

Number of moles (kmol)

600

q = 0.85 Solution interact. NH3,1000 µm

500

r po

va

or vap

400

no ventilation

300

ventilation HE-limit

200

liqu

liq

id

uid

liq

100 0

uid

0

600 Number of moles (kmol)

vapor

40

80 120 Time (s) (b)

q = 0.85 Dry air NH3, 100 µm

500

160

200

vapor

400 no ventilation ventilation

300

HE-limit

200 liquid

100 0

0

40

80 120 Time (s)

160

200

(c) FIGURE 27.7 Number of moles of ammonia vapor and liquid versus time predicted by AERCLOUD, for 85% liquid releases in dry air and in 99.99% humid air allowing for ammonia–water interactions. The initial droplet radius ranges from 1000 ␮m to 100 ␮m. The curves have been computed using three model options: including and excluding droplet ventilation and in the homogeneous equilibrium limit.

27.13

27.14

CHAPTER TWENTY-SEVEN

Number of moles (kmol)

600

q = 0.85 Solution interact. NH3,100 µm

500

vapour

400

no ventilation ventilation

300

HE-limit

200 100 0

liquid

0

40

80 120 Time (s)

160

200

(d) FIGURE 27.7 Number of moles of ammonia vapor and liquid versus time predicted by AERCLOUD, for 85% liquid releases in dry air and in 99.99% humid air allowing for ammonia–water interactions. The initial droplet radius ranges from 1000 ␮m to 100 ␮m. The curves have been computed using three model options: including and excluding droplet ventilation and in the homogeneous equilibrium limit. (Continued )

quilibrium computations enhances the mass and heat transfer. In the HE limit these transfer rates are the largest possible and ammonia is therefore vaporizing more quickly. The effect of ambient air moisture is again to increase the rate of vaporization at small times and suppress it at larger times. For the initial droplet radii of 100 ␮m, the differences of model predictions of the three model options are less than 1% for both the cases considered. For a droplet radius of 10 ␮m, the curves are practically identical, implying that thermodynamic equilibrium prevails at all times. We can conclude that the HE model gives good results concerning contaminant vaporization for the selected cases if the droplet radius is not larger than about 100 ␮m. It can also be shown by simple qualitative arguments that the HE model is better applicable for small droplets compared to larger ones in the continuum regime (Kukkonen et al., 1994). Figure 27.8 shows the relative difference of droplet vaporization times predicted by the aerosol model, compared to the HE limit, for ammonia–water interaction cases with 85% and 30% liquid initially. The vaporization time has been defined here as the time by which 80% of the liquid ammonia within the droplet has been vaporized. In accordance with the earlier results, the differences of the model predictions and the influence of droplet ventilation increase with the droplet size.

27.5

CONCLUSIONS The homogeneous equilibrium model is often adopted in dispersion models of two-phase jets and clouds, but it has not previously undergone a critical test. We have therefore tested the homogeneous equilibrium model, in the context of the dispersion of a dense gas cloud, against a more sophisticated aerosol model. The thermodynamical nonequilibrium effects in a two-phase cloud have two essential consequences: (1) the thermodynamical behavior of the mixture is different, in particular the temperature and density evolution, and (2) the deposition of contaminant liquid may cause

MODELING ATMOSPHERIC DISPERSION

27.15

Difference of models

5 No ventilation Ventilation

4 3 2

23 se ca

1 0

cas

0

200

e 21

case 23 case 21

400 600 800 Initial droplet radius (µm)

1000

FIGURE 27.8 Relative difference of AERCLOUD predictions compared to the homogeneous equilibrium limit, i.e., (tam ⫺ the)/ the, where t is the droplet vaporization time. The subscript ‘‘am’’ refers to the aerosol model, with or without the convective mass and heat fluxes, and ‘‘he’’ refers to the homogeneous equilibrium limit.

a decrease of concentration. Our prime objective here was to examine the vaporization and condensation processes, and we have therefore ignored deposition. However, the deposition of droplets is certainly an important process in some conditions; clearly, deposition is highly sensitive to droplet size. We find that if the droplets are small enough, then the heat and mass transfer processes to the droplet are fast enough (compared to the overall dispersion rate) to make homogeneous equilibrium a good approximation. In the heavy gas dispersion tests done here, 100 ␮m droplets are seen to be small enough. The question of the adequacy of the homogeneous equilibrium model is thus reduced to that of estimating droplet sizes. It is clear that in many processes involving violent flashing of pressure-liquefied gases, the drop sizes will be expected to be small enough to admit homogeneous equilibrium as a good approximation in the subsequent advection and dispersion of a heavy cloud.

27.6

REFERENCES Blackmore, D., M. Herman, and J. Woodward. 1982. ‘‘Heavy Gas Dispersion Models,’’ Journal of Hazardous Materials, vol. 6, pp. 107–128. Bricard, P., and L. Friedel. 1998. ‘‘Two-phase Jet Dispersion,’’ Journal of Hazardous Materials, vol. 59, pp. 287–310. Brighton P. W. M. 1985. ‘‘Area Averaged Concentrations, Height Scales, and Mass Balances,’’ Journal of Hazardous Materials, vol. 11, pp. 189–208. Britter, R. E. Recent Research on the Dispersion of Hazardous Materials, European Communities Report EUR 18198 EN, Luxembourg. Carissimo B., S. F. Jagger, N. C. Daish, A. Halford, S. Selmer-Olsen, K. Riikonen, J. M. Perroux, J. Wu¨ rtz, J. Bartzis, N. J. Duijm, K. Ham, M. Schatzmann, and R. Hall. 1999. ‘‘The SMEDIS Database and Validation Exercises,’’ presented at SMEDIS Workshop on the 6th International Conference on Harmonisation within Atmospheric Dispersion Modeling for Regulatory Purposes, Rouen, October 11– 14. Daish, N. C., R. E. Britter, P. F. Linden, S. F. Jagger, and B. Carissimo. 1998. ‘‘SMEDIS: Scientific Model Evaluation of Dense Gas Dispersion Models,’’ in Proceedings of the 5th International Conference

27.16

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on Harmonisation within Atmospheric Dispersion Modelling for Regulatory Purposes, May 18–21, Rhodes, Greece, National Centre for Scientific Research ‘‘Demokritos,’’ Athens, pp. 54–61. Hanna, S. R., and P. J. Drivas. 1987. Vapor Cloud Dispersion Models, Center for the Chemical Process Safety of the American Institute of Chemical Engineers, New York. Hewitt, G. F., and M. J. Pattison. 1992. Modeling of Release and Flow of Two-phase Jets. The Safe Handling of Pressure Liquefied Gases (IBC), London, November 26–27. Johnson, D. W. 1991. ‘‘Prediction of Aerosol Formation from the Release of Pressurized, Superheated Liquids to the Atmosphere,’’ in Proceedings of International Conference and Workshop on Modeling and Mitigating the Consequences of Accidental Releases of Hazardous Materials, New Orleans, May 20–24, American Institute of Chemical Engineers, New York, pp. 1–34. Koopman, R. P., T. G. McRae, H. C. Goldwire, Jr., D. L. Ermak, and E. J. Kansa. 1986. ‘‘Results of Recent Large-Scale NH3 and N2O4 Dispersion Experiments,’’ in Heavy Gas and Risk Assessment III: Proceedings of Third Symposium on Heavy Gas and Risk Assessment, Bonn, November 12–13, ed. S. Hartwig, D. Reidel, Dordrecht, pp. 137–156. Koopman, R. P., D. L. Ermak, and S. T. Chan. 1989. ‘‘A Review of Recent Field Tests and Mathematical Modelling of Atmospheric Dispersion of Large Spills of Denser-than-Air Gases,’’ Atmospheric Environment, vol. 23, pp. 731–745. Kukkonen, J. 1990. Modelling Source Terms for the Atmospheric Dispersion of Hazardous Substances, Commentationes Physico-Mathematicae 115, The Finnish Society of Sciences and Letters, Helsinki. Kukkonen, J., T. Vesala, and M. Kulmala. 1989. ‘‘The Interdependence of Evaporation and Settling for Airborne Freely Falling Droplets,’’ ‘Journal of Aerosol Science, vol. 20, no. 7, pp. 749–763. Kukkonen, J., M. Kulmala, J. Nikmo, T. Vesala, D. M. Webber, and T. Wren. 1993. Aerosol Cloud Dispersion and the Suitability of the Homogeneous Equilibrium Approximation, AEA Report AEA / CS / HSE R 1003 / R, Warrington, Cheshire. Kukkonen, J., M. Kulmala, J. Nikmo, T. Vesala, D. M. Webber, and T. Wren. 1994. ‘‘The Homogeneous Equilibrium Approximation in Models of Aerosol Cloud Dispersion,’’ Atmospheric Environment, vol. 28, pp. 2763–2776. Kulmala, M., and T. Vesala. 1991. ‘‘Condensation in the Continuum Regime,’’ Journal of Aerosol Science, vol. 20, pp. 337–346. Kulmala, M., T. Vesala, and P. E. Wagner. 1993. ‘‘An Analytical Expression for the Rate of Binary Condensational Particle Growth,’’ Proceedings of the Royal Society of London A, vol. 441, pp. 589– 605. Marshall, V. C. 1987. Major Chemical Hazards, Ellis Horwood, New York. McQuaid, J., and B. Roebuck. 1985. Large Scale Field Trials on Dense Vapor Dispersion, Final Report on the Heavy Gas Dispersion Trials at Thorney Island 1982–1984, Report EUR 10029, Commission of the European Communities, Brussels, pp. 417. Moodie, K., and B. C. R. Ewan. 1990. ‘‘Jets Discharging to the Atmosphere,’’ Journal of Loss Prevention in the Process Industries, vol. 3, pp. 68–76. Muralidhar, R., G. R. Jersey, F. J. Krambeck, and S. Sundaresan. 1995. ‘‘A Two-Phase Release Model for Quantifying Risk Reduction for Modified HF Alkylation Catalysts,’’ Journal of Hazardous Materials, vol. 44, pp. 141–183. Nielsen, M., S. Ott, H. E. Jørgensen, R. Bengtsson, K. Nyre´ n, S. Winter, D. Ride, and C. Jones. 1997. ‘‘Field Experiments with Dispersion of Pressure Liquified Ammonia,’’ Journal of Hazardous Materials, vol. 56, pp. 59–105. Nikmo, J., J. Kukkonen, T. Vesala, and M. Kulmala. 1994. ‘‘A Model for Mass and Heat Transfer in an Aerosol Cloud,’’ Journal of Hazardous Materials, vol. 38, pp. 293–311. Nolan, P. F., G. N. Pettitt, N. R. Hardy, and R. J. Bettis. 1990. ‘‘Release Conditions Following Loss of Containment,’’ Journal of Loss Prevention in the Process Industries, vol. 3, pp. 97–103. Schmidli, J., S. Banerjee, and G. Yadigaroglu. 1990. ‘‘Effects of Vapour / Aerosol and Pool Formation on Rupture of Vessels Containing Superheated Liquid,’’ Journal of Loss Prevention in the Process Industries, vol. 3, pp. 104–111. Vesala, T. 1991. ‘‘Binary Droplet Evaporation and Condensation as Phenomenological Processes,’’ Ph.D. thesis, University of Helsinki; Commentationes Physico-Mathematicae 127, The Finnish Society of Sciences and Letters, Helsinki.

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