PGS2-1997 Methods for the Calculation of Physical Effects Chapter 4 Vapour Cloud Dispersion

7/29/2019 PGS2-1997 Methods for the Calculation of Physical Effects Chapter 4 Vapour Cloud Dispersion

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4.1

Chapter 4Vapour cloud dispersionE.A. Bakkum, N.J. Duijm


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CPR 14EChapter 4 of the Yellow Book

4.3

Modifications to Chapter 4 (Vapour Cloud Dispersion)with respect to the first print (1997)

Numerous modifications were made concerning typographical errors. A list is givenbelow for the pages on which errors have been corrected.

Some figures in this chapter were poorly printed; viz. figures 4.5, 4.16 and 4.18 aswell as the scheme in section 4.5.1. These prints have been improved.

Equation (4.51) on page 4.70 is a general expression for a continuous release.

In section 4.5.3.3 the equations (4.60) and (4.61) describe short duration releasesinstead of semi-continuous releases.In equation (4.60b)

x

(x) has been replaced by

x

(u

a

t).Below equation (4.61c) a sentence has been added concerning linear interpolation on

x

for values of x less than 50 m.In section 4.5.3.6 it has been added that the proposed simplifications will introduceconsiderable continuity problems in calculating the concentration.

In section 4.5.4.1 describing the free turbulent jet model, equation (4.71) has beenreplaced and (4.73) has been removed. Hence all subsequent equations in chapter 4have been renumbered.

The constant in equation (4.79) has been adjusted, as well as the empirical constantsC

u

and C

c

in table 4.10.

The dispersion parameters were changed into the values of the 2

nd

edition ofCPR14E.

In section 4.5.4.2 the equations (4.83) and (4.84) have been adjusted, as well as theconstants in equations (4.87) and (4.92).In equation (4.149) an ending bracket just before the division line has been added.

In section 4.6.3 concerning application of the Gaussian plume model, the examples

have been updated.Now subsection 4.6.3.1 describes the calculation of a plume and subsection 4.6.3.2shows the calculation of explosive mass of an instantaneous puff.Hence table 4.14 and figures 4.19 up to and including 4.21 have been replaced.

In subsection 4.6.4.1 the example on the free jet model has been corrected and figure4.22 has been replaced.

Section 4.6.5 describes example calculations for four source term submodels in theSLAB model. Some errors in the SLAB model have been corrected. All calculationsare updated and figures 4.23 up till and including 4.25 are replaced, whereas figure4.27 has been added. Furthermore the values for the Concentration averaging time

and Stability class in table 4.23 have been adjusted.


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4.4


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4.5

List of symbols Chapter 4

cross-flow area of cloud or plume (4.26) m2

b half-width radius of jet or plume (4.16) mbo source radius (section 4.3.5.1.) mbox, boy, boz source half dimensions in down-wind,

cross-wind and vertical direction,respectively (4.62) m

bx, by half-width of cloud in down-wind andcross-wind direction, respectively (4.114) m

by,m effective plume half-width includingmeandering (4.145) m

bz cloud height (4.23) m

C general constants, to be defined in the textc(...) concentration as a function of the parts per unit volume

terms between brackets (4.5) or kg m-3

co concentration at source (4.25) p. p. u. v. or kg m-3

cc concentration at jet, plume or cloudcentre-line (4.28) p. p. u. v. or kg m-3

cr, cg peak concentration at maximum plume heightand plume touch-down, respectively (4.85) p. p. u. v. or kg m-3

cmax maximum concentration (Section 4.3.5.1) p. p. u. v. or kg m-3

cmean time-averaged concentration (Section 4.3.5.1) p. p. u. v. or kg m-3

cp specific heat at constant pressure (4.1) J kg-1 K-1

cp,a ambient specific heat at constantpressure (4.115) J kg-1 K-1

clfl, cufl lower and upper flammability concentration,respectively (4.66) p. p. u. v. or kg m-3

d particle diameter (4.4) m

e albedo (4.37) -

Egh ground heat (4.115) J m-1 s-1Epc phase change energy (4.115) J m

-1 s-1

f ( ) general function, to be defined in the text,of the terms between brackets

f Coriolis parameter (4.8) s-1

fx, fy, fz drag term in down-wind, cross-wind andvertical direction, respectively kg s-2

F buoyancy flux factor (4.20) m4 s-3

Fo buoyancy flux factor at source (4.21) m4 s-3Fd dry deposition flux (4.5) kg m

-2 s-1

Fn wet deposition flux (4.6) kg m-2 s-1


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4.6

Fx(..), Fy(..), Fz(..)expressions which account for along-wind, lateral,and vertical dispersion, respectively (4.28) m-1

Fr Froude number (4.17) -

g acceleration of gravity (4.1) m s-2

g' effective gravity g.( - a)/a (4.20) m s-2

go' effective gravity at source g.(o - a)/a(section 4.3.5.1.) m s-2

h height of plume or cloud centre-line (4.11) mhi mixing height (4.9) m

hs source height (4.86) mhr final plume rise (4.84) mhB plume rise due to buoyancy (4.88) mhM plume rise due to momentum (4.92) m

H0 sensible heat flux (4.1) J m-2 s-1

H r net radiation heat flux (4.37) J m-2 s-1

H rs incoming solar radiation (4.36) J m-2 s-1

H ri isothermal net radiation heat flux (4.41) J m-2 s-1

Hg ground heat flux (4.39) J m-2 s-1

H l latent heat flux (4.39) J m-2s-1

H heat of condensation (4.144b) J kg-1

I rain intensity (4.6) mm per hour

K reaction coefficient (4.3) -ks transfer coefficient (4.45) -

L Monin-Obukhov length(stability parameter) (4.1) m

L s constant mLb buoyancy length-scale (4.101) m

m mass fraction of concentration (4.107) kg kg-1

M o momentum from source (4.17) kg m s-2

N cloud cover (4.36) -

p (partial) pressure (4.142) N m-2

pa atmospheric pressure (4.143) N m-2

ps saturation pressure (4.143) N m-2

q mass flow rate (4.11) kg s-1

Q released mass (4.52) kg

r radial distance (4.23) m


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4.7

R gas constant (4.142) J mol-1

K-1

s coordinate along jet or plumecentre-line (4.16) m

sl length-scale parameter (4.71) mslfl distance to LFL (4.80) msh maximum jet height (4.79) mss length of potential core (Table 4.10) mso distance between real source and

virtual source (Table 4.10) m

t time (4.12) s

tav averaging-time (4.15) stmin minimum averaging-time (4.131) stL integral time-scale of atmospheric

turbulence (4.13) sti integral time-scale of dispersion (4.54) str release duration (4.60) stpk time to peak concentration (4.155) s

T temperature (4.138) KT a ambient temperature

(at screen height, about 2 m) (4.1) KT r upper level ambient

temperature (50 m) (4.41) K T o ambient temperature at

roughness height (4.40) K T s surface or sea temperature (4.40) KT w wet bulb temperature

(at screen height) (4.45) K

T * turbulent temperature scale (4.34) K

u (down-wind) velocity of dispersing material (4.18) m s-1

ua ambient wind velocity (4.11) m s-1

uc

velocity at jet or plume centre-line (4.73) m s-1

uo velocity at source (4.17) m s-1

uy, uz lateral and vertical velocity of dispersingmaterial, respectively (4.18) m s-1

u* surface friction velocity (4.1) m s-1

vo initial volume flow rate (4.100) m3 s-1

ve entrainment term (4.159) m3 s-1

V cloud volume (4.25) m3

Vo initial volume (4.25) m3

ws sedimentation velocity (4.4) m s-1

wd deposition velocity (4.5) m s-1

we entrainment velocity (4.16) m s-1


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4.8

we,t top entrainment velocity (4.23) m s-1

we,e edge entrainment velocity (4.23) m s-1

we,x down-wind entrainment velocity (4.161) m s-1

we,y cross-wind entrainment velocity (4.161) m s-1

wH effective heat transfer velocity (4.136) m s-1

x down-wind horizontal coordinate m

xb shape parameter (4.168) mxc down-wind distance to centre of

mass of cloud (4.156) mxlfl, xufl down-wind distance to lower and upper

flammability level, respectively (4.66) mxu upwind extension of adense plume (4.101) m

xr distance to maximum plume rise (4.83) mxg distance to plume touch-down (4.86) mxv, xvy, xvz virtual distances to account for finite source

dimensions during evaluation of plume size(4.68) m

y cross-wind horizontal coordinate (4.2) mylfl, yufl cross-wind distance to lower and upper

flammability limit, respectively (4.65) my

bshape parameter (4.147) m

z vertical (upward) coordinate (4.2) mzb shape parameter (4.147) mzo surface roughness length (4.7) mzs constant (4.47) mzlfl height to lower flammability limit (4.65) m

Greek symbols

moisture availability constant (4.38) -l vapour mass fraction coefficient (4.138) -

ratio of specific heat of air to latentheat of water divided by rate ofchange (derivative) of saturation specifichumidity with temperature (4.38) -

l liquid mass fraction coefficient (4.138) -

d dry adiabatic lapse rate (4.33) 0.011 K m-1

von Karman constant (4.1) 0.4

wash-out coefficient (4.6) s-1 per (mm per hour)

a molar weight of dry air (4.108) kg mol-1

s molar weight of dispersing material (4.108) kg mol-1


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4.9

moist molar weight of moist air (4.139) kg mol-1

kinematic viscosity (4.4) m2 s-1

density (4.2) kg m-3

a ambient density (4.1) kg m-3

p density of particle (4.4) kg m-3

s density of dispersing material (4.141) kg m-3

Stefan Boltzmann constant (4.37) 5.67.10-8 W m-2 K-4

x, y, z down-wind, cross-wind and vertical

dispersion parameters of cloud (4.11) myi dispersion parameter of instantaneous

plume width (4.56) mv, w standard deviation of turbulent velocities

in cross-wind and vertical direction,respectively (4.13) m s-1

conserved quantity (4.18) arbitrary

earths latitude (4.8) N

solar elevation (4.36)

earths longitude (appendix 1) W

stability function (4.32) -

earths rotational speed (4.8) 7.27 . 10-5 s-1

Note: the numbers between brackets refer to equations.


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4.10


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4.11

Glossary of terms

adiabatic lapse rate vertical temperature gradient in an atmospherewhere the potential temperature gradient is zero

aerodynamic diameter diameter of a spherical particle that has the samefree-fall velocity as the arbitrarily shaped particle

aerosol airborne particle

albedo fraction of solar radiation that is reflected into

space

ambient surrounding atmosphere

atmospheric stability the extent to which vertical temperature(= density) gradients promote or suppressturbulence in the atmosphere

Boussinesq approximation approximation in the Navier Stokes equationwhich considers density differences to berelevant for buoyancy only and not formomentum terms

buoyancy the upward force (Archimedes force) that iscaused by a cloud or plume in which the densityis lower than the surrounding atmosphere

continuous release release during a long time with a constantcontaminant mass flow rate

dense gas gas which has a higher specific weight than thesurrounding ambient air

density specific weight

deposition absorption of gas or particles by the ground orvegetation

dispersion mixing and spreading of gases in air, whichcauses clouds to grow

dispersion coefficient standard deviation of the concentration profile ina cloud in one direction (lateral, i.e. cross-wind,vertical along-wind)

eddy random vortex motion in turbulence


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4.12

eddy diffusivity apparent diffusivity due to turbulence, whichacts like a molecular diffusivity in diffusing massor momentum

ensemble set of dispersion situations which are describedby nominally the same initial and boundaryconditions. Differences which occur within anensemble are due to (turbulent) randomness

entrainment mixing of (clean) air into a cloud or plume

flammable limits concentration which is either the LFL or UFL

friction velocity by definition the cube root from (minus) theshear stress at the surface

gravity spreading the horizontal spreading of a dense gas cloud onthe ground due to the hydrostatic force which isa result of the density difference

integral time-scale time-scale which can be calculated by integrating(over time lag) the autocorrelation function of aturbulent quantity (i.e. velocity)

instantaneous release release during which in a (very) short time a(large) amount of gas is released

Karman constant proportionality constant appearing in therelation between velocity gradient and shearstress for turbulent flow near a rough surface

kinematic viscosity viscosity divided by fluid density

LFL lower flammability limit, below thisconcentration too little flammable gas is presentin the air to maintain combustion

mixed layer layer below the mixing height

mixing height height of the turbulent boundary layer over theground

Monin-Obukhov length length-scale which characterises the atmosphericstability

Navier Stokes equations momentum balance equations for movements influids

neutral atmosphere atmosphere where potential temperature isconstant with height


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4.13

partial pressure fraction of total pressure due to the presence of agas; total pressure is the sum of all partialpressures of the gases present in a mixture

passive dispersion dispersion solely caused by atmosphericturbulence

phenomenological models models which present an empirical relationbetween initial and boundary conditions andobserved phenomena, without tempting todescribe the physical relationships

precipitation rain, snow, etc.

pure jet jet which is only driven by momentum

pure plume plume which is only driven by buoyancy

plume rise the ascent of a plume in the air due to verticalmomentum at the source or buoyancy

puff cloud spreading in all directions due to aninstantaneous release

regula falsi method to find a root of a function

roughness length artificial length-scale appearing in relationsdescribing the wind speed over a surface, andwhich characterises the roughness of the surface.Note the sizes of the elements causing theroughness can be more than ten times larger thanthe roughness length

saturation pressure pressure of a gas at saturated state. It depends ontemperature only; above the saturation pressurethe gas will condense

saturation specific humidity specific humidity of air with saturated watervapour, a function of temperature and, to a lesserextent, total pressure

sedimentation the deposition of particles on the ground due totheir free-fall velocity


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4.14

semi-continuous release with a constant contaminant mass flowrate during a finite time

shear vertical velocity gradient

shear stress stress exerted by the wind on the ground surfacedue to friction

specific humidity humidity expressed as mass fraction of watervapour in air

stable atmosphere where potential temperature

increases with height

still air air without detectable wind speed or direction

turbulence random motions in a fluid due to instabilities of a large-scale flow

two-phase flow flow in which two phases are present, e.g. gas andliquid droplets or liquid with gas bubbles

UFL upper flammability limit, above thisconcentration too little oxygen is available tomaintain combustion

unstable atmosphere where potential temperaturedecreases with height

wash-out absorption of gas or particles in rain-drops fallingthrough a gas cloud or plume


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4.15

Table of contents Chapter 4

Modifications to Chapter 4, Vapour Cloud Dispersion .......................3

List of symbols Chapter 4......................................................................5

Glossary of terms.................................................................................11

4 Vapour cloud dispersion......................................................................174.1 Introduction..................................................................................174.2 The phenomenon of dispersion......................................................19

4.2.1 Introduction to Section 4.2..................................................194.2.2 Atmospheric turbulence and dispersion................................194.2.3 Atmospheric stability...........................................................214.2.4 Passive dispersion................................................................234.2.5 Jets and plume rise..............................................................244.2.6 Dense gas dispersion............................................................244.2.7 Some other aspects of dispersion..........................................25

4.3 General overview of existing models...............................................294.3.1 Introduction to Section 4.3..................................................294.3.2 Models to describe atmospheric stability and atmospheric

parameters..........................................................................294.3.3 Models to describe passive dispersion...................................32

4.3.4 The modelling of turbulent jets and plumes..........................354.3.5 The modelling of dense gas dispersion..................................42

4.4 Selection of models.......................................................................524.4.1 Introduction to Section 4.4..................................................524.4.2 Schemes for atmospheric stability categories and

meteorological data..............................................................524.4.3 Selection of models for passive dispersion.............................534.4.4 Selection of models for jets and plumes................................544.4.5 Selection of dense gas dispersion models ..............................55

4.5 Description of models ...................................................................574.5.1 Introduction to Section 4.5..................................................574.5.2 Models to determine atmospheric stability and

vertical variation of wind speed ............................................594.5.3 Passive dispersion................................................................694.5.4 Models for jets, plumes and plume rise.................................794.5.5 Dense gas dispersion models................................................86

4.6 Application of selected models: calculation examples....................1114.6.1 Introduction to Section 4.6................................................1114.6.2 Examples of stability calculations.......................................1114.6.3 Application of the Gaussian plume model...........................1134.6.4 The application of models for jets and plumes....................1184.6.5 The application of dense gas dispersion models ..................121


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4.16

4.7 Discussion.................................................................................. 1304.7.1 Introduction to Section 4.7............................................... 1304.7.2 Changes in atmospheric conditions.................................... 1304.7.3 Still air ............................................................................. 1304.7.4 Accuracy of the Gaussian plume model.............................. 1304.7.5 Dispersion over obstacles.................................................. 1314.7.6 Concentration fluctuations................................................ 1314.7.7 Removal processes............................................................ 132

4.8 Literature................................................................................... 133

Appendix 4.1 A method for estimating the solar elevation


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4.17

4 Vapour cloud dispersion

4.1 Introduction

Hazardous material that is released in the atmosphere will be transportedand diluted by the wind. This chapter will provide models that enable the predictionof this transportation and dilution process for various release scenarios andconditions. The output of these models are concentrations (as a function of time) atany location surrounding the release (up to several kilometres). Moreover, the totalamount of hazardous material in a flammable vapour cloud between the upper and

lower flammability limits will be predicted.

Dispersion will be restricted to dispersion within the atmospheric mixed layer in astationary state. T his means that down-wind distances are restricted to typically10 km.

The input of the models described herein will be the results of the source term model,described in chapters 2 (Outflow and spray release) and 3 (Evaporation). Once thecloud, plume or jet interacts with the ambient air the dispersion processes describedin this chapter are to be applied.

The Sections 4.2 through 4.5 in this chapter will each address all of the four main

themes of this chapter viz. atmospheric stability; passive dispersion; jets, plumes, andplume rise; and dense gas dispersion. However, each section will treat the topics witha different aim:

Section 4.2 provides the essential and basic understanding of the phenomena ofinterest. I t provides information on atmospheric turbulence in general, and addressesshortly some topics beyond the main themes, viz. deposition, chemical reactions andobstacle effects.

Section 4.3 provides a general overview of the currently available methods andmodels for the main themes, which enable quantified analysis. It includes somephysical/mathematical background to models. T he aim of the section is also to

position the models selected for detailed description among other existing models.

Section 4.4 presents the considerations which led to the selection of models which arefound to be acceptable for use and which are selected for detailed description.

Section 4.5 provides the detailed and complete description of the selected models.Whenever calculations or analyses have to be made, all necessary information can befound in this chapter.It is recommended to read the considerations presented insection 4.7 concerning validity and applicability of the models, before stating thecalculations.

Section 4.6 gives examples of the use of the models presented in section 4.5.


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4.18

Finally, section 4.7 provides some information on general restrictions on the use ofmodels. I t includes in some cases suggestions regarding methods to overcome theserestrictions or it recommends alternative approaches.


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4.19

4.2 The phenomenon of dispersion

4.2.1 Introduction to Section 4.2

This section provides the essential and basic understanding of thephenomenon of dispersion. Section 4.2.2 introduces atmospheric turbulence and dispersion. Section 4.2.3 explains atmospheric stability and the important parameters that

describe stability. Section 4.2.4 describes the origin and effects of jets and plume rise.

Section 4.2.6 describes the basic phenomenon of dense gas dispersion.

Section 4.2.7 discusses some topics related to dispersion, viz. heat transfer,chemical reaction, deposition and effects of obstacles.

4.2.2 Atmospheric turbulence and dispersion

The lower part of the atmosphere in which the releases take place is calledthe mixed layer. The height of the mixed layer (the mixing height) varies mostlybetween 200 and 2000 m. T he wind flow in this mixed layer is almost alwaysturbulent, i.e. the movement of the air fluctuates continuously in velocity anddirection due to the presence of eddies. T hese turbulent eddies are very effective in

dispersing material through the air, a factor of at least 1000 more effective thanmolecular diffusion.

Turbulent flows are generally described in statistical terms, i.e. by a time-averagedmean velocity and a time-averaged standard deviation of velocity fluctuations aroundthis mean value.

Even for still-air, stable conditions turbulence is present due to temperaturedifferences in the atmosphere which cause movements of air.According to Hanna [1990], the standard deviations of horizontal turbulent velocitiesare not lower than 0.5 m/s (with a scatter of 0.3 m/s).

Turbulence is generated in two ways viz.:

1. Mechanically through the resistance of the earths surface on the wind. Thiscauses a decrease of wind speed to the surface (velocity gradient or shear). Thisshear causes turbulence due to flow instabilities, and a downward turbulent fluxof momentum to compensate the resistance force. T he wind shear depends mainlyon the upper wind speed and the surface roughness.

2. Thermally due to heating of the surface (mainly by the sun). Hot patches of airnear the surface start to rise. T his causes an upward turbulent flux of heat.

Without any generation of turbulence, turbulence will die due to viscous dissipation.Much more important in the atmosphere is the suppression of turbulence in steadily


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4.20

stratified layers of air, i.e. in layers in which temperature increases upwards. Verticalmovements of air are suppressed.

An important aspect of atmospheric turbulence is the large range of sizes of theturbulent eddies, from several hundreds of metres diameter down to millimetres. Thedifferent sizes of eddies act differently in diffusing a puff of gas, see Figure 4.1.Turbulent eddies smaller than the size of the puff will uniformly disperse material andincrease the size of the puff (Figure 4.1a). T urbulent eddies much larger than thesize of the puff will only displace the puff without changing its size or geometry(Figure 4.1b). Eddies comparable to the size of the puff will change its geometry andincrease its contour (Figure 4.1c). T hese aspects are also relevant to theunderstanding of concentration fluctuations. At a certain position one will observefluctuations of concentrations due to meandering of the whole puff, and due todeformation of the puff. T he concentration fluctuations are suppressed by smalleddies within the puff.As the puff grows, it is apparent that the number of eddies contributing to meanderingwill decrease, whereas sub-puff eddies will contribute more and more to dispersion.The nature of turbulent dispersion changes during the life-time of the puff.

Figure 4.1 Dispersion of a puff of material under three turbulence conditions:

(a) Puff embedded in a field in which the turbulent eddies are smaller than thepuff.

(b) Puff embedded in a field in which the turbulent eddies are larger than the puff.(c) Puff embedded in a field in which the turbulent eddies are comparable in sizeto the puff [Seinfeld, 1986]

(a)

(b)

(c)


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4.21

The concepts are also applicable to continuous releases of gas (or aerosol). Forcontinuous releases there is a problem of defining an appropriate averaging-time fordetermining concentration and plume size. An instantaneous picture of a continuousrelease will show the effect of individual large eddies displacing the plume centre-line,see Figure 4.2, the instantaneous concentration distribution. I f one observes thesubsequent action of a number of eddies, the time-averaged concentrationdistribution will show a Gaussian distribution (Figure 4.2, the 10-min or 1-houraveraged concentration distribution). I f there is a maximum size of turbulent eddies,one will be able to define an averaging-time beyond which no change of the averagedconcentration distribution can be observed. This is the case for eddies in the verticalplane, as the eddies need to be smaller than the mixing height. However, in thehorizontal direction no such limit exists: the wind direction will change continuously.This means that the horizontal cross-wind size of a plume will be ever increasing withincreasing averaging-time. T his implicates that for any model for prediction ofconcentration an indication of averaging-time is necessary. The specification of anaveraging-time is also relevant from a point of view of consequences: someconsequences occur at very small-time-scales, almost instantaneously, (e.g. vapourcloud combustion) at time-scales of human inhalation (toxicity) up to tens of minutes(deposition at ground).

Figure 4.2 Plume boundaries (left) and concentration distributions (right) of a plume at

different averaging-times [Seinfeld, 1986]

The inherent variability of turbulence and turbulent diffusion should be kept in mindwhen assessing model predictions of atmospheric dispersion. In fact, models (try to)predict an ensemble-average puff or plume, i.e. the averaged puff or plume from alarge number of releases under nominally similar conditions.

4.2.3 Atmospheric stability

When an airpatch moves from the surface upwards it will expand aspressure decreases. Due to expansion the temperature of the airpatch will decrease.The temperature of a parcel which has been brought adiabatically to ground level is

called potential temperature. When the airpatch has the same temperature as itssurroundings during its travel upwards, the atmospheric stability is referred to asneutral: no forces due to density differences (originating from temperature

y

x

1 hr average boundary

10 min average boundary

instantaneous boundary

Mean wind direction =Time mean axis of plume

Relative concentration

10 min averagedistribution

instantaneousdistribution

1 hour averaged distribution


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4.22

differences) are exerted on the patch. I f the temperature of the airpatch becomeslower than its surroundings, the atmosphere is stable; the airpatch is forceddownward. If on the contrary the temperature becomes higher than its surroundings,the atmosphere is unstable: the airpatch will accellerate upwards.

The vertical gradients of temperature are illustrated in F igure 4.3. From thermo-dynamics one calculates the dry adiabatic lapse rate to be -0.01 K /m. For saturatedair, the condensation of water vapour reduces the cooling of the air and the decreaseof temperature with height is less. T his is the wet adiabatic lapse rate. For surfacetemperatures of about 290 K, the wet adiabatic lapse rate is about -0.005 K/m. (Dueto the change of saturation pressure, the difference between the dry and wet adiabaticlapse rate is larger at higher temperatures.) For temperature gradients between thetwo, the atmosphere is stable for dry air and unstable for saturated air. T his regime iscalled conditionally unstable.

Figure 4.3 Temperature gradients and stability regimes in the atmosphere

During unstable conditions there is a heat flux from the surface upwards (this occursas the surface is heated by the sun). During stable condition the heat flux isdownwards (the surface is cooled at night by heat radiation to the sky). The stabilityof the mixed layer is determined by the ratio of turbulence generated by thetemperature gradient and the turbulence generated mechanically by wind shear at thesurface. T his ratio can be expressed by a characteristic length-scale, the Monin-Obukhov length, which is defined as:

600

500

400

300

200

100

0283 284 285 286 287 288 289 290 291 292 293

temperature (K)

unstable

dry adiabaticlapse rate

wet adiabaticlapse rate

conditionallyunstable

stableheigh

t(m)

L ac

pT

au

*

3

gH

o

----------------------= (4.1)(m)


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4.23

The friction velocity u

*

is by definition the square root of the shear stress divided bythe density of air at the surface. H

o

is the sensible heat flux (see Section 4.5.2). In(4.1)

a

, c

p

, and T are the density, specific heat, and near surface temperature of theair, respectively.

is the so-called Von K arman constant (

0.4) and g thegravitational acceleration.The Monin-Obukhov length provides a measure of stability of the mixed layer. L canbe interpreted as the height above the ground where turbulence generated by windshear equals the turbulence dissipated by the heat flux. In unstable conditions thereis not such an equilibrium, and this length is negative:

L > 0 stable (H

o

< 0)L < 0 unstable (H

o

> 0)L =

neutral (H

o

= 0)

Table 4.1 Interpretation of the Monin-Obukhov length L with respect to atmospheric

stability

Qualitative schemes to characterise stability are often used, e.g. the Pasquill scheme,which rates from class A (unstable) through D (neutral) to F (stable). Methods todetermine atmospheric stability and the relation with the Pasquill scheme aredescribed in Section 4.3.2.

4.2.4 Passive dispersion

Situations where the dispersion of a puff or cloud of material is governedsolely by the atmospheric turbulence are called passive dispersion. The state of theatmosphere is not changed by the presence of the material in the air.Assuming homogeneous turbulence and wind speed (i.e. the turbulence and windspeed are the same at all locations in the air) one can derive that the concentrationdistribution of an initially small puff of material becomes Gaussian in shape. Inpractice it appears that one observes a Gaussian distribution of concentration in verymany occasions. M any dispersion models are based on this Gaussian distribution;these will be discussed in more detail in Section 4.3.3.

L Stability condition

Small negative - 100 m < L < 0 Very unstable

Large negative - 10

5

m < L < - 100 m Unstable

Very large (positive or negative) |L| > 10

5

m Neutral

Large positive 10 m < L < 10

5

m Stable

Small positive 0 < L < 10 m Very stable


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4.24

4.2.5 Jets and plume rise

4.2.5.1 Jets

Gases which are released with high velocity will cause jets. As long as thevelocity inside the jet is high compared to velocities in the ambient air, the extent andmixing in the jet is only affected by the properties of the jet itself.The velocity difference between the jet and the (assumed quiescent) surrounding airgenerates fine scale turbulence which causes the jet to spread sideways. The velocityin the jet reduces (about inversely proportional to the distance to the point of release).Finally the velocity will be reduced to such extent that passive dispersion takes over.

Models for free jets in still air and jets in cross-flow (wind) are discussed in Sections4.3.4.1 and 4.3.4.2 respectively.

4.2.5.2 Plume rise

Material released in the atmosphere may rise because:1. the material is less dense (buoyant) compared to the surrounding air and/or2. the material contains upward momentum.

Theoretically, in a neutrally or unstably stratified atmosphere, the material will riseindefinitely. However, due to mixing, the volume over which buoyancy and/ormomentum is distributed increases, and this causes the rate of plume rise to decreaserapidly. When vertical motion of the plume is of the same magnitude as the turbulentmotion of the plume one assumes the plume or puff to be at its final height.

In case the atmosphere is steadily stratified, there is a height at which the releasedmaterial, taking into account mixing during plume rise, will be in equilibrium with thedensity of the air at that height, which then is assumed to present the final plume rise.

After the plume has reached final plume rise, dispersion may be assumed to be

passive.For continuous releases from small source areas (stacks, vent pipes) a large numberof models for plume rise exist. T hese will be discussed in Section 4.3.4.2 to 4.3.4.3.

4.2.6 Dense gas dispersion

Releases of material denser than the ambient air introduce some specialeffects which affect the dispersion.The released material will descent to the surface and, once on the ground, spreadradially under influence of the gravitational forces. This self-induced flow produces ashallow cloud with increased horizontal extent. At the front of the so-called gravity-current a head will develop with a strong vorticity (see Figure 4.4). T his velocity fieldis deterministic in nature and will replace, for the duration of the gravity spreading,


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4.25

the random atmospheric turbulence. The self-induced flow will increase mixing,especially just behind the gravity head.

After gravity spreading has finished, the vertical variation of density in the cloud willcause a stable stratification in the cloud which reduces dispersion in vertical direction.Finally the effects of density will be dispersed and negligible and dispersion willbecome passive.

Dense gas dispersion models will be discussed in Section 4.3.5.

Figure 4.4 Gravity spreading of a dense gas cloud

4.2.7 Some other aspects of dispersion

In this section some aspects of dispersion are discussed that will not bediscussed separately or in detail in the remainder of this chapter. These are: heattransfer, chemical reactions, deposition, and obstacle effects.

4.2.7.1 Heat transfer and change of buoyancy

Often the total buoyancy in a cloud, either negative (dense cloud) orpositive, remains constant during the dispersion process. T his is referred to by saying

buoyancy is conserved.For an instantaneous release this can be expressed as:

Here,

(x,y,z) denotes the density in the cloud and

a

(z) denotes the density of theambient air at the same height.

head head

air

vortex

gravityspreading

x,y,z( ) a z( ){ }dxdydz cons ttan=all space

(4.2)(kg)


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4.26

However, some processes in the cloud may change the total buoyancy:

heat transfer from the surface to the cloud;

vaporisation of liquid aerosol in the cloud;

condensation and evaporation of water vapour aerosol in the cloud;

chemical reactions which change the number of moles and/or which areendothermic or exothermic.

Heat transfer from the surface is relevant to cold plumes, originating e.g. fromreleases of cryogenic gases. T he warm surface heats the cold gas, which can changefrom negatively buoyant to positively buoyant, causing the cloud to lift from theground.Depending on the velocity of the cloud over the surface and the temperaturedifference, heat transfer is dominated by forced convection or by free convection. Forforced convection, the heat flux from the surface is about proportional to the frictionvelocity and the temperature difference. For free convection, the heat flux isindependent of velocity and about proportional to the temperature difference to thepower 4/3 rd.

Vaporisation of liquid aerosol, during releases of e.g. pressurised liquified gases, willcool down the air/gas mixture in the cloud, increasing its negative buoyancy.

The ambient air contains always some water vapour. For releases of cold material, thewater vapour mixed into the cloud may condensate, thereby increasing the buoyancy.At a later stage, the condensed water may well evaporate again, cooling down thecloud, but these effects often happen at a distance or travel time from the release,where density effects are no longer relevant.

If the released material reacts with components in the air during which the numberof molecules change, the total buoyancy is no longer conserved. I f the number ofmolecules decreases, the cloud will become denser and vice versa. T he heat producedor required by the chemical reaction will change the total buoyancy as is the case withcondensation or evaporation of water.

4.2.7.2 Chemical reactions

Chemical reactions will also convert the initialy released substance intoanother secondary substance. From a point of view of hazards one might be interestedin the primary substance or in the secondary.In general, a reaction of the form

A + B

C + D

is controlled by a reaction constant K which determines equilibrium between thesubstances A, B, C and D:

K

A[ ] B[ ]

C[ ] D[ ]------------------ (Kcanbezeroorinfinite)= (4.3)(-)


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4.27

([A], [B], etc, denote concentrations of the substances A, B, etc.)

Local deviations from equilibrium are due to finite reaction rates and, relevant nearthe release, to concentration fluctuations inside the cloud.

4.2.7.3 Deposition

Material can be removed from the cloud by deposition of material on theground. One may discriminate the following deposition mechanisms:

sedimentation of aerosols;

dry deposition of gases and aerosols at the surface;

wet deposition by precipitation (rain, snow) of gases and aerosols from the cloud.

All particles in the air have a finite sedimentation velocity. The sedimentation velocityw

s

can be approximated by Stokes law:

Here d is the (aerodynamic) diameter of the particle and the specific density of theparticle,

a

the density of air, g is gravitational acceleration, and

is the kinematic

viscosity of air.In practice one may assume that the sedimentation velocity of particles smaller than10 diameter (w

s

< 0.02 m/s) is negligible, and that dispersion of these particles issimilar to dispersion of gas.

Dry deposition of aerosols and gases can be described as a deposition flux F

d

whichis the product of a deposition velocity w

d

and a concentration c at some referenceheight above the surface.

F

d

= w

d

c (kg m

-2

s

-1

) (4.5)

The deposition velocity is the inverse of the sum of three resistances; viz. anaerodynamic resistance dominated by turbulence and stability of the atmospherebetween the surface and the reference height; a resistance which is caused by the flowaround structures at the surface (surface induced resistance), dominated by thefriction velocity and surface roughness, and a surface resistance, which depends onthe properties of the substance (e.g. solubility in water) and the type of surface(vegetation).The aerodynamic resistance can change from about 1000 s/m (stable conditions,smooth surface and low wind speed) down to 3 (high-wind speed, rough surface).Typical values for the surface induced resistance are in the order of 10 s/m. Surfaceresistances can vary from order of 10 to more than 1000 s/m.

Wet deposition only occurs during precipitation, but it can be much more

effective than dry deposition. Wet deposition can be described by a wash-out

ws1

18-----d

2g

p

a--------= (4.4)(m s

-1)

p

m


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4.28

coefficient (s

-1

per mm precipitation per hour)

1)

and an intensity of precipitation I(mm/hour)

1)

. T he wet deposition flux F

n

can be written as:

Here, c(z) is the concentration in the cloud. For aerosol is typically 4.10

-4

s

-1

fora precipitation of 1 mm per hour.

4.2.7.4 Obstacle effects

Dispersion over flat terrain with homogeneous roughness of which theindividual roughness elements are smaller than the height of the cloud is rather wellunderstood. T he flow and dispersion around individual obstacles or arrays ofobstacles is difficult to describe and to quantify in a general way (for a number ofspecific situations approximate solutions have been found).Obstacle effects however are important. A rule of thumb to estimate when to accountfor obstacles is that the smaller value of height or width (in cross-wind direction) islarger than 0.5 to 1. times the local cloud height.Behind obstacles a so-called recirculation zone exists. T his zone may extend to about

10 times the obstacle height [Duijm and Webber, 1993]. Due to increased turbulencein the wake of the obstacle, the maximum groundlevel concentration down-wind ofthe recirculation is lower than in the absence of the obstacle. Nearer to obstacles nogeneral trend can be indicated: an increase of concentration is possible and increasesby a factor of 2 have been reported.

1)

and I are usually expressed in these non-SI units.

Fn I c z( )dz

surface

cloudheight

= (4.6)(kg m-2 s-1)

I


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4.29

4.3 General overview of existing models

4.3.1 Introduction to Section 4.3

Section 4.3 provides an overview of methods and models used fordispersion calculations. I t presents alternative methods and provides shortdescriptions of the contents of the main types of models in a general way.In this section the following themes are discussed:

Section 4.3.2 presents atmospheric stability schemes and methods to predictcharacteristics of the atmosphere.

Section 4.3.3 presents passive dispersion models with emphasis on the Gaussianplume models.

Section 4.3.4 presents models for jets and plumes in still air and in ambient wind,and simple models for plume rise.

Section 4.3.5 presents various types of dense gas dispersion models.

4.3.2 Models to describe atmospheric stability and atmosphericparameters

4.3.2.1 Atmospheric stability classifications

All models used to describe or predict atmospheric dispersion needinformation about the stability of the atmosphere, mainly to define the rate at whichthe material is passively dispersed by atmospheric turbulence. T his information canbe obtained through various methods:

Direct measurement of the intensity of wind direction fluctuations in lateral andvertical direction [Erbrink, 1991]. This method requires measurements which arenot routinely carried out. The link with alternative schemes (e.g. the Pasquillscheme, see below) is difficult. However, the results can be directly applied tocalculate the dispersion parameters (plume widths) for the Gaussian plumemodel.

Classification of horizontal wind direction traces as defined by Singer and Smith

71953], the so-called Brookhaven Gustiness Classes, changing from Class A(unstable) to Class D (stable). I t is not clear whether this scheme can representstability adequately over different types of terrain and different climates.

Classification of atmospheric stability from routine meteorological data, such asthe well-known Pasquill scheme [Pasquill and Smith, 1983] which is based onobservations of wind speed, cloud cover and time of day. The Pasquill scheme iscurrently the most applied scheme.

Direct measurement of friction velocity u

*

and Monin-Obukhov lenght L, whichprovides an unambiguous measure of stability in the lower part of the atmosphericboundary layer (see Section 4.2.2). These measurements are not routinely carriedout. The link to e.g. the Pasquill scheme can be made.


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4.30

Calculation of friction velocity and M onin-Obukhov length from routinemeteorological data and terrain specification as by Holtslag, Nieuwstadt and VanUlden [Holtslag, 1984; van Ulden and Holtslag, 1985; Holtslag and Nieuwstadt,1986; Holtslag and de Bruin, 1987; Holtslag, 1987]. T he link to the Pasquillscheme can be made.

Most widely-used dispersion models make use of the Pasquill stability categories(A-unstable to F-stable) and parametrisations of the local lateral and vertical plumewidth (

y

and

z

) corresponding to Pasquill-Gifford, Briggs, Smith, etc. (see e.g.Seinfeld [1986], pp 577).

However, better results can be obtained if the qualitative schemes are exchanged in

favour of schemes using quantifications of physical parameters of the boundary layer(such as roughness length z

o

, heat flux H

o

, friction velocity u

*

, and Monin-Obukhovlength L).A complete scheme using routinely meteorological data has been developed byHoltslag, Nieuwstad and Van Ulden. T he scheme is based on providing an estimateof the surface heat budget, i.e.:

sensible heat flux + latent heat flux + heat flux into the soil =

net radiation =

(1 - albedo) * solar radiation + incoming long wave radiation

- outgoing long wave radiation

The sensible heat flux determines the stability of the atmosphere. Holtslags schemeprovides methods of estimating all terms in the equation above based on routinelyavailable data and information of the local situation (e.g. surface roughness, albedoand solar elevation). Holtslags scheme is described in Section 4.5.2.

It is possible to link the Pasquill stability scheme to the Monin-Obukhov lengthdepending on local surface roughness by a graph proposed by Golder [1972]. T hismethod is included in Section 4.5.2 (see Figure 4.11).

The stability over extended water surfaces (seas) depends mainly on the temperaturedifferences between the air and the water. From the temperature difference one cancompute the sensible heat flux which, together with the friction velocity, determinesstability. Hsu [1992] developed a simple scheme (nomogram) to relate the stability interms of Monin-Obukhov length and Pasquill classes to the temperature differencebetween the water surface and the air and the wind speed, which are routinelyavailable data, see Section 4.5.2.4 and Figure 4.12.


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4.31

4.3.2.2 Other atmospheric characteristics

Almost all characteristics of the mixing layer can be expressed in terms ofstability parameters (friction velocity u* and Monin-Obukhov length L). Quantitiesrelevant to dispersion are: vertical variation of wind speed; vertical variation of standard deviations of turbulent velocities in cross-wind and

vertical direction (v and w respectively); mixing height.

Variation of wind speedThe vertical variation of wind speed is often approximated by a power law.

Alternatively, wind speed ua can be described by relations which follow fromsimilarity theory in the surface layer:

ua(z) = f(z/zo, z/L) (m s-1) (4.7)

Here f(z/zo, z/L) is an empirical function, u* the friction velocity and the VonKarman constant. T his relation is adequate up to a height of about 100 m abovewhich the vertical variation of wind speed can be neglected. Well-known suggestionsfor the function f are from Businger and Dyer and Bradley and can be found in thetext books Pasquill and Smith [1983], Seinfeld [1986], Panofsky and Dutton [1984],and Stull [1988]. More details can be found in Section 4.5.2.

Turbulence parametersValues for standard deviations of turbulent velocities in the surface layer are providedby Panofsky and Dutton [1984]. Proposals for formulae describing the verticalvariation of these quantities can be found in Gryning et al. [1987], Seinfeld [1986]and Stull [1988]. M ore information is provided in Sections 4.4.2 and 4.5.2.

Mixing heightThe mixing height depends on: height of inversion layers; surface friction velocity; Monin-Obukhov length; latitude of position on earth.

The height of inversion layers is very often dominated by advection of large-scale airmasses in which these inversion layers are present. These have little relation with thelocal situation of the mixing layer at any given moment. In The Netherlands themixing height is dominated by these inversion layers about one third of the time. Thissituation will be different in other areas, depending on location on earth, proximity toseas and oceans, etc.

If no inversion layers are present, one might estimate the local mixing height from u*,L and the Coriolis parameter f, which is defined as:

f = 2 sin (s-1) (4.8)

u*-----


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4.32

with the earth rotation (7.27 . 10-5

s-1

) and the latitude of position on earth (about51 for T he Netherlands).For purely neutral conditions, the mixing height can be calculated as (see e.g.Panofsky and Dutton, [1984]):

For stable conditions one may write for stationary conditions [Panofsky and Dutton,1984, Pasquill and Smith, 1983]:

No stationary solution exists for the mixing height in unstable conditions. The mixingheight at a certain time of day can be calculated from the vertical temperature profileat sunrise and the time integrated heat flux since sunrise.

Statistical information about mixing heights can also be estimated frommeteorological data. T his information accounts for the effects of elevated inversions,but it can not be applied to all positions on earth. For calculations involving unstableconditions without information of time, this is the only applicable method.Suggestions for mixing height predictions, based on statistical meteorologicalinformation and formulae (4.9) and (4.10) above, are presented in Section 4.5.2.

4.3.3 Models to describe passive dispersion

Passive dispersion is governed by atmospheric turbulence. Atmosphericturbulence is determined by the stability of the atmosphere and the height above thesurface. Gryning et al. [1987] have defined scaling regions in the atmosphere(Figure 4.5).

Gaussian plume models have been used extensively for all regions mentioned in

Figure 4.5. T he basic expression for the Gaussian plume model for a continuousrelease is:

hi 0.2u* f= (4.9)(m)

hi 0.4 u*

f-----L= (4.10)(m)

c x,y,z( )q

2uayz------------------------ .exp

y

2

2

y

2

--------

.exph z ( )

2

2

y

2

-----------------

= (4.11)(kg m-3)


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4.33

Figure 4.5 The scaling regions of the atmospheric boundary layer shown as function of the

dimensionless height z/h

i

, and the stability parameter h

i

/L. A detailed discussion

can be found in Holtslag and Nieuwstadt [1986]. When used to determine

dispersion regions, the dimensionless height is replaced by h/h

i

where h is the

source height.

The equivalent expression for an instantaneous release is written as:

In (4.11) and (4.12) c(x,y,z) is the concentration at position x,y,z. Q and q are thetotal released mass and release rate, respectively. u

a

is the ambient velocity at whichthe plume or puff is advected by the wind.

x

,

y

, and

z

are the so-called dispersionparameters in along-wind, cross-wind, and vertical direcion, respectively. The heightof the plume centre-line is denoted by h.For plumes near the ground or near the mixing height additional terms appear in(4.11) and (4.12) to account for reflection of material against the surface or themixing height.

Within the framework of the Gaussian plume models, differences may appear in:

the choice of the plume averaged transport velocity u

a

: and

the choice of the dispersion parameters

x

,

y

, and

z

.

In most applications of the Gaussian plume model the transport velocity is taken asthe wind speed at plume height h, with a minimum of h for surface releases.

Entrainment layer

Mixed layer Near neutralupper layer

z-less scaling layer

Intermittencylayer

Localscaling layer

Surface layerSurface layer

Freeconvectionlayer

1.2

0.8

zhi

0.1

0.01

-100 hiL

-10

unstable

-1

neutral

0 1 5 10hiL

N.neu.up.layer

stable

c x,y,z,t( )Q

2( )3/2

xyz--------------------------------.exp

x u a

.t ( )2

2

x

2

-----------------------

.expy2

2

y

2

--------

.=

exph z( )2

2z2

-----------------

(4.12)(kg m-3)


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4.34

With respect to the dispersion parameters

x

,

y

, and

z

different approaches can befollowed, two of which are mentioned here. T he fundamentals of the differentapproaches are discussed in Pasquill and Smith [1983].

The first approach is based on the statistical theory of dispersion in homogeneousturbulence. T his leads to expressions in which the spread (i.e. the dispersionparameter) in any direction is proportional to the intensity of velocity fluctuations inthat direction times a function of travel time t and some integral time-scale of theturbulence t

L

:

x

=

u

f (t,t

L

);

y

=

v

f (t,t

L

);

z

=

w

f (t,t

L

) (m) (4.13)

Several proposals for the function f (t,t

L) for y and z have been suggested, see e.g.Pasquill and Smith [1983], Panofsky and Dutton [1984], Stern et al. [1984], andSeinfeld [1986]. The function as suggested by Draxler [Stern et al., 1984] for y andz is widely used.

The other approach is to provide fits to experimental data without a link to theoreticalconsiderations. This leads to expressions of the form:

x = f (x); y = f (x); z = f (x) (m) (4.14)

Widely-used suggestions for the function f(x) for y and z are those of Pasquill-Gifford, M.E. Smith and Briggs [Pasquill and Smith, 1983; Stern et al., 1984;

Panofsky and Dutton, 1984]. Seinfeld [1986] also includes the suggestions by Klugas incorporated in German guidelines for stack plume dispersion modelling.

Strictly speaking, the Gaussian plume model can not be used for description ofpassive dispersion in the surface layer because the vertical variations of wind speedand turbulence intensity in the surface layer cause the vertical concentrationdistribution to deviate from a normal distribution. Based on similarity theory and theconcept of eddy-diffusivity one is able to provide analytical solutions orapproximations for the vertical concentration distribution [Pasquill and Smith, 1983;Van Ulden, 1991, Gryning et al., 1987].

L ittle information is available on the along-wind dispersion parameter x for (near)

instantaneous releases. Not only turbulence in along-wind direction contributes tothe increase of x, but also the vertical variation of wind speed contributes to anincrease ofx. Van Ulden [1991] provides a scheme for x which accounts for botheffects.

Schemes to estimate the cross-wind dispersion parameter y need to indicate theaveraging-time for which they are valid. In order to transfer y-values to otheraveraging-times a power law expression with an exponent between 0.1 and 0.5 isgenerally accepted [Guinnup, 1992]. An exponent of 0.2 is widely used [Ermak,1990];

y,t1 y,t2

tav 1,

tav 2,---------

0.2

= (4.15)(m)


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4.35

Here y,t1 and y,t2 refer to a horizontal dispersion parameters for an averaging-timeof tav,1 and tav,2, respectively.

Relations for cross-wind spread of instantaneous releases are provided by Pasquill andSmith [1983], Panofsky and Dutton [1984] and Van Ulden [1991]. These should beused as minimum values for short-term exposure.

The selection of methods for description in the Yellow Book is justified in Section4.4.3. The description of selected methods is provided in Section 4.5.3.

4.3.4 The modelling of turbulent jets and plumes

4.3.4.1 Jets and plumes in still air

The formalism for the modelling of free turbulent jets has been developedalready several decades ago. It is the aim of this section to present the backgroundinformation of the jet model, which is needed in gas dispersion calculations. T heplume model, which is similar in many aspects, is also discussed. T he aim of therecent scientific work has not been to make further improvements to the theory, butis concerned mainly with the collection of experimental data to determine theempirical constants in the formalism. T he monograph of Chen and Rodi [1980] on

the subject contains an outstanding and extensive compilation of the experimentalresults. T heir recommended values of model parameters are discussed in Section4.5.4.1.

Within the context of the Yellow Book, the definition of a pure jet is: a source ofmomentum and energy in the atmospheric environment. The fluid motion in a jetrelease is governed by the inertial forces. When the density of the released compoundis different from the density of air, the buoyancy force will compete with the inertialforce. The jet is called a positively buoyant jet when the inertial force and thebuoyancy force act in the same direction; for a negatively buoyant jet the directionsare opposite. Examples of buoyant jet releases are releases of vent stacks, exhausts,safety valves and punctures in pipes and reservoirs. All positively buoyant jets

eventually become plumes. T he jet momentum is continuously increased bybuoyancy, and at some point sufficiently far above the source the momentum of thejet is largely independent of the initial momentum. This represents the fundamentalproperty of a (pure) plume: it is a source of buoyancy. The pure plume can alsoemerge from a source release without initial momentum. e.g. heat diffusers, coolingtowers, chimneys, burners, pool fires and evaporating liquid spills.

The analysis of jets in this section is made under the following simplifyingassumptions: The release of the chemical compound consists of a turbulent outflow of gases.

Two-phase releases are discussed in chapter 2 of this book. The source is axially symmetric, with a uniform outflow. These conditions provide

a good approximation for the description of a wide class of controlled or accidentalgas releases.


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4.36

It is assumed that buoyancy can either be neglected, or the buoyancy force isparallel to the inertial forces. In the latter case the release is in vertical direction,since the buoyancy force is caused by gravitation.

The analysis is only valid when there are no large density differences between thejet and the surrounding atmosphere. This is called the Boussinesq approximation.The restriction is not severe, since jets and plumes which violate the Boussinesqapproximation soon mix by entrainment of the surrounding air.

The atmospheric environment consists of quiescent air at ambient conditions.The environment is neutrally stratified. It does not contain walls or otherobstacles.

Compressibility effects are neglected. In situations where the chemical compoundis stored under high pressure a source model must be used to describe the initialexpansion of the pressurised release (see Chapter 2), before using the jet model.

A turbulent jet (or plume) consists of 3 regions (see Figure 4.6): an initial region, atransition region, and the fully developed jet. T he initial region consists of the coreflow and a surrounding shear layer, which forms the boundary with the quiescent air.For a uniform exit velocity the core flow is nearly free of shear. In the transitionregion, the jet entrains air by the turbulent eddies in the shear layer, and the detailsof the exit core flow become obliterated. The resultant eddy-dominated flow is calledfully developed.

Figure 4.6 The development of a turbulent jet in still air

Non-buoyant jetsThe analysis in this section aims to describe the distributions of velocity and

concentration in the third region of a fully developed jet. Since the jet develops freeof externally applied constraints, the distribution functions are dynamically similar ifall quantities are nondimensionalised by local length and time-scales. I t is convenient

fully

developedregion

transitionregion

initialregion

shearlayer

core

2 bo


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4.37

to relate length and time-scales to the axial distance s from the origin of the jet and tothe exit velocity uo. In this approach it is assumed that the origin of the jet is a virtualpoint source instead of a finite circular source. I n other words, the dimension of thesource is neglected. T he virtual point source does not necessarily coincide with thereal source. However, experimental studies have shown that the separation distanceis small, so s is approximately the distance from the real source exit.

The mass flow q(s) in the jet increases with the axial distance:

dq/ds = 2 b a we (kg m-1 s-1) (4.16).

b is an estimate of the radius between the jet centre-line and the boundary surface of

the jet; a is the ambient density. Equation 4.16 defines the entrainment velocity we.The physical interpretation is that an air flow passes through the boundary surface ofthe jet with a velocity we. A dimensional analysis shows that we is proportional to thejet centre-line velocity uc. T he ratio of these velocities is a measure of the dilution ofthe jet and is called the entrainment coefficient. T he dynamic similarity of the non-buoyant jet in the region of fully developed flow has several other implications for thephysical properties: The decay of the centre-line velocity uc and of the centre-line concentration cc of

the chemical compound are both proportional to the inverse of the axial distance s. In lateral direction, perpendicular to the jet centre-line, the velocity and the

concentration are only a function of the non-dimensional coordinate y/s, where yis the radial distance from the centre-line. T hus, the width b of the jet isproportional to the axial distance s, and the jet shape is conical with the origin, forwhich b is equal to zero, located at the virtual point source.

Note that the concentration of the compound is defined in parts per unit volumerelative to the atmospheric level, i.e. it is the difference between the local value andthe value in air.

Buoyant jetsThe physical properties of the non-buoyant jet are determined by the cross-sectionintegrated axial momentum M o and by the ambient density a. For the pure, non-buoyant jet the density ratio o/a of the initial chemical compound and ambient airis unity. In the description of a buoyant jet, the buoyancy flux factor Fo of thechemical compound and ambient air appears as a third parameter. I t can be shown[Chen and Rodi, 1980] that dynamic similarity can not exist for buoyant jets, so it isnot possible to derive simple relations for the flow quantities, as in the case of the non-buoyant jet. In situations where the buoyant term in the momentum equations of thejet are of minor importance, the non-buoyant relations are approximately valid. Thevalidity depends on the axial distance s, since for large s the buoyant force alwaysbecomes competitive with the inertial force. A negatively buoyant jet will rise to amaximum height under its inital momentum and then fall back on itself. A positivelybuoyant jet will gradually change into a pure plume. T he properties of a pure plumeare discussed below. Chen and Rodi define in their monograph length-scales thatdetermine the regions of non-buoyant and pure plume behaviour.


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4.38

Pure plumesA plume is a buoyant jet whose initial momentum is nearly zero. For convenience itis assumed in this text that: The plume weighs less per unit volume than air, so the plume is driven upward

under the pressure of the surrounding air. Molecular diffusion is negligible compared with turbulent transport.It can be shown [Chen and Rodi, 1980] that with these assumptions the pure plumeis self-preserving, or in other words, dynamically similar. The flow quantities can bederived by performing a dimensional analysis, which begins with the definition of theFroude number:

Here, uo is the exit velocity from the source, bo is the radius of the real axial jet source,o is the density of the released material and a is the ambient density. It can be shownthat the centre-line velocity uc and the centre-line concentration cc of the chemicalcompound are dependent on the Froude number and decay, respectively, as s-1/3 ands-5/3. Chen and Rodi have compared the simple relations of non-buoyant jets and pureplumes with experimental data, and determined the values of the model parametersin the equations. T his is discussed in section 4.5.4.1.

4.3.4.2 Jets and plumes in cross-wind

The properties of jet and plume releases into a quiescent atmosphere havebeen discussed in the previous section. In the applications in industry andenvironmental problems, the release is usually deflected under the action of ambientwind. This configuration, which is called a jet in a cross-flow, is addressed in thepresent section. The dispersion of the chemical compound in the jet can be calculatedwith models which solve the full set of conservation equations of the relevant physicalquantities. The review of Ooms and Duijm [1984] on the modelling of the dispersionof dense gas stack plumes has not lost its actuality and is used as the basis for thepresent general discussion of the formalism. A list of the most prominent models isgiven in this section, and one of these models is treated in detail in Section 4.5.4.2.

An easier, although less accurate, way to calculate the plume dispersion is to apply thesimple Gaussian plume model. In this case a correction must be made for the initialrise of the released compound due to momentum and density effects. Severalmathematical expressions have been derived by Briggs [1969] to model the height ofthe total plume rise. T he present section discusses the mathemathics of the model,and in doing so follows an approach that was originally adopted by Davidson [1989].Section 4.5.4.3 gives recommendations on the application of Briggs formulae.

The analysis of jets and plumes in a cross-flow starts from the same simplifyingassumptions as given in the introduction of Section 4.3.4.1, with two modifications: The atmosphere is not quiescent, but the ambient air is moving as a uniform

horizontal turbulent flow.

In the discussion of integral trajectory models, the Boussinesq approximation isnot used.

Fro

a o------------------

uo2

2bog-----------= (4.17)(-)


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4.39

Section 4.3.4.1 was concerned with free jets and plumes, and their dispersion in anunbounded atmosphere. In practical situations of gas cloud dispersion the importantquantity is usually the concentration of the chemical compound at ground level,which can only be determined with a model that accounts for the height of the sourcewith respect to the ground.

The wind imposes a pressure field on the jet, which deflects the jet, deforms theinitially circular jet cross-section into a kidney shape and establishes a pair of counter-rotating vortices in the cross-section. Entrainment of the air from the wind into thejet causes further deflection and after some distance the jet movement isapproximately horizontal, with a mean velocity approaching the wind velocity (seeFigure 4.7).

Figure 4.7 The plume trajectory in a cross-flow

One-dimensional integral trajectory modelsIntegral methods are based on simple profile shapes (mostly tophat or Gaussian) forthe velocity, temperature and density distributions in the jet cross-section. Althoughsome models attempt to describe the vortex structures in the jet with an ellipsoid, itis assumed here that the jet cross-section remains circular along its path. Theconservation equations for mass, momentum, energy and chemical compound canthen be integrated over the jet cross-section to yield ordinary differential equations of

some characteristic velocity and density. T o close the system of equations, it is

fully developed region

bent-over, Boussinesq plume

transitionregion

initialregion

wind

plume axis

plume centre-line

2 bo

cross-section


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4.40

necessary to make further assumptions about the entrainment of air into the jet. T heintegral equations are of the form:

In the equation s is the axial coordinate of the jet or plume, and y is the radialcoordinate, perpendicular to the axis. is the density and u the flow velocity in thejet, b is the width parameter of the jet and a is the density of ambient air. The termwe(s) can be interpreted as the entrainment velocity (cf. equation 4.16) and containscontributions from 3 processes:

Jet entrainment occurs due to the shear stress with the atmosphere. Air is entrained by the velocity component of ambient wind, perpendicular to thejet axis.

Entrainment due to atmospheric turbulence can be neglected close to the stack,but not at larger distances where it becomes the dominant process.

The functions and a, represent the conserved physical quantity in the jet and inambient air. I t can be equal to 1 (unity), or the horizontal velocity ux, or theconcentration c or the heat content cpT , leading to the conservation equations of,respectively, the total mass, the horizontal momentum, the chemical compound massand the thermal energy. The function f(s) is non-zero only in the momentumequation, where it represents a drag term which takes into account changes inmomentum due to the pressure field. I t is a point of controversy whether the inclusion

of drag forces in the momentum equation is really necessary. The set of equations iscompleted with the conservation equation for vertical momentum, in which case equals uz, and the right-hand side of equation 4.18 includes the influence of thegravitational force.

In the equation the stratification of the atmosphere is not specified. T he simplestplume trajectory models introduce two additional assumptions: The distributions of physical quantities within the plume are constant (tophat

shape) at any given distance from the source. The atmosphere is not stratified.The simplifications allow for an integration of the conservation equations over thewhole plume cross-section. T he integrated version of equation 4.18 is:

Since by definition T a= 0 in an unstratified atmosphere, the integrated conservationequation of thermal energy implies that the buoyancy flux factor F is a conservedquantity, see also Section 4.2.7.1 and formula (4.2). I t is a measure of the densitydifference between the jet and ambient air, and defined as:

d

ds----- u 2y dy

o

2

b

2b aa we s( ) f s( )+= (4.18).kgm1 s 1( )

d

ds----- ub

2[ ] 2baawe s( ) f s( )+= (4.19)kgm

1 s

1 ( )

F g' ub2 .= (4.20)(m4 s-3 )


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4.41

The density difference is contained in the effective gravity g'.

The conservation equations are usually solved numerically, and yield the averageddensity

and velocity u of material in the plume, the tilt

(=arctan(u

z

/u

x

)) of theplume axis, and the width b of the plume, as a function of the distance s along theplume axis. T he tophat profile of the physical quantities in the plume can, if required,afterwards be transformed into a Gaussian shape.

The differences in the assumptions of integral trajectory models become apparentwhen a comparison is made between the models, which have been implemented infour current dense gas dispersion models, i.e. DEGADIS, HGSYSTEM, PHASTand SLAB. The four implemented plume models are, respectively:

the Ooms model (1972),

the HFPL UME model (M cFarlane et al. 1990),

the TECJET model (Havens and Spicer, 1990),

the model of Hoot, Meroney and Peterka (HMP, 1973).

The HMP differs from the other models in some aspects:

It consists of analytic expressions for the ground locations of maximal plume riseand touch-down, the maximal height and the centre-line concentrations at theselocations. T he other models must be solved by means of numerical techniques;

Entrainment from atmospheric turbulence is not included. Therefore, it does notdescribe passive gas releases.

In T able 1, information of the models is presented on a number of physical aspects.

Table 4.2 Physical aspects of 1-dimensional integral plume models

Plume rise formulae

For many practical problems, it is sufficient to estimate the trajectory of the jet orplume in a cross-flow from empirical correlations. T he derivation of the relevantparameters is best done by considering each of the terms in the conservationequations. In this section, the approach of Davidson [1989] is followed. F irst, theequations are simplified further by application of the Boussinesq approximation(

=

a

= constant, except in density differences) and the bent-over plume

assumption (u = ua). These assumptions allow to derive an analytical equation for theplume trajectory, which includes the effects of both inertial forces and buoyancyforces.

cross

section

cross-

wind

profile

surface

roughness

entrainment

from

atmospheric

turbulence

drag

force

zone of flow

establishment

slumped

plume

interface

Ooms ellips Gaussian yes yes yes yes no

HFPLUME circle tophat yes yes no no yes

TECJET circle Gaussian no yes yes yes no

1)

1)

Included in dispersion model.

HMP circle tophat no no no no no

1)


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4.42

Since the Boussinesq approximation and the bent-over plume approximation areused in the derivation of the trajectory equation, it is not intended for the calculationof properties close to the source. T he plume rise due to initial momentum M

o

andinitial buoyancy F

o

contribute with a down-wind dependency of x

1/3

and x

2/3

,respectively. This implies that at distances further away from the source, the buoyantplume rise is the dominating term.

The formalism of Davidson is incomplete in the sense that it does not treat the effectsof atmospheric stratification and stability. As a consequence, the equations predict aninfinite plume rise, whereas in reality it is observed that the plume levels off at acertain height, after which it is influenced almost solely by the atmosphericconditions.

Assuming that the transition to a horizontal plume occurs at a down-wind distance x

r

and that the plume rise due to the initial momentum is relatively small, the finalplume rise is equal to

h

r

= constant F

o1/3

/ u

a

(m) (4.21).

In equation 4.21, F

o

is the buoyancy flux factor of the source (defined by 4.20 withu

o

, g

o

' and b

o

). The constant has dimension (m

2/3

). It has been shown by Briggs[1969] that experimental observations of plume rise from tall stacks can be describedby equation 4.21, provided that the constant is a function of F and also of theatmospheric stability. Briggs has also derived expressions for momentum-dominated

plume rise, of the form

Here the constant has dimension (m

2/3

s

-2/3

).Many models of final plume rise have been proposed by Briggs and by other authors,and most of them assume the functional behaviour which is given in the aboveequations. T hey differ mainly in the way that the effects of atmospheric stability aretreated. A well-established model is presented in section 4.5.4.3

4.3.5 The modelling of dense gas dispersion

Our interest is in catastrophic releases of a chemical compound, with a largevolume or volume flow rate. Many hazardous volatile materials are heavier than air,for one or more of the following reasons:

The molecular weight of the chemical compound is higher than that of air.Examples are: LPG, cyclohexane, freon and chlorine.

The chemical compound is released as a cold gas, or cooled due to the evaporationprocess. Examples are: LNG, ammonia and hydrogen fluoride.

In case of a flashing release, the cloud is a mixture of aerosols and vapour. T his isoften seen in ammonia releases.

Chemical reactions occur which are endothermic or result in the pick-up of

atmospheric water vapour. Reactive chemical compounds are ammonia, hydrogenfluoride and nitrogen tetroxide.

hr cons t/ua2/3tan = (4.22).(m)


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4.43

The temporal behaviour of the dispersion process is determined by the failure mode:

In an instantaneous release, a volume of gas is released in a very short time.Examples of instantaneous sources are severely ruptured vessels or rapidlyemptying pressurised tanks.

In a time-varying release the volume flow continues over a long period in anirregular way. An example is a liquid pool which is both spreading andvapourising.

A continuous release is characterised by a constant volume flow rate with asufficiently long duration for a steady state to be formed. Examples of this sourcetype are steady-state liquid pools and small ruptures in pipes and vessels.

The instantaneous release type differs from the other two in that along-winddispersion plays an important role in the dilution of the cloud.During the dispersion process of a dense (negatively buoyant) gas cloud, 4consecutive phases are observed:

The initial phase. The fluid motion is a strong function of the release conditions.

The gravity spreading phase. T he entrainment is mainly due to the turbulencegeneration by the gravity-induced motion of the cloud.

An intermediate phase. Entrainment is due to an interplay between theatmospheric turbulence and gravitational forces.

The passive (neutrally buoyant) dispersion phase.The gravity spreading phase (see also Section 4.2.6) provides a good illustration ofthe effects which are typical for dense gas:

The density gradient induces gravity spreading in the horizontal direction. Thesegravity currents drive the flow within the cloud, independent of the atmosphericwind.

M ixing is primarily through the self-generated vortices at the edge of the cloud(also called gravity-front).

M ixing by atmospheric turbulence at the top of the cloud is suppressed by thedensity gradient.

Turbulence within the cloud is reduced due to the stable stratification (i.e. anegative vertical density gradient) of the dense gas layer.

As a consequence, a dense gas cloud remains in the lower part of the atmosphere, witha large spread in the lateral direction. Meandering due to atmospheric turbulence isless than for passive clouds. Attempting to describe dense gases by adapting Gaussianmodels suitable for passive clouds is inherently inadequate, because the Gaussianmodels do not account for the momentum and energy effects.Present dense gas dispersion models can be roughly divided into 3 classes:

Phenomenological models, in which the dispersion behaviour is described by aseries of nomograms.

Intermediate models, such as box models and

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PGS2-1997 Methods for the Calculation of Physical Effects Chapter 4 Vapour Cloud Dispersion