Model of dilution of forced liquid

8/7/2019 Model of dilution of forced liquid

1/36

A Model of Dilution of Vapor- Liquid Chemical Plume in

Horizontal Wind.

CHAPTER 1

1


2/36

INTRODUCTION

An accidental release of liquid or gaseous fuel indoors or to the outside

is a safety problem that is frequently considered by the chemical and waste

remediation industries that process and/or store flammable substances. The

release almost always occurs in the form of a negatively or positively

buoyant, continuous jet which mixes with the ambient air. All possible

fuel/ambient air mixture compositions occur within the buoyant jet between

its source and the far field of the jet.There are many cases m the chemical

industry where various chemicals are stored under pressure, and at

temperatures exceeding their normal boiling point In the case of an

accidental release under such conditions, the material emerges from the

release point (e.g., orifice, rupture, etc.) as a liquid that immediately flashes

to form a two-phase jet.

After the extremely serious accidents which happened in the chemical,

petrochemical and petroleum industry in the last 3 decades, the safety

authorities have to reconsider the procedures and the models dedicated tothe assessment of potential hazards. In particular, one may have to

determine the impact on the environment of an accidental release of toxic or

flammable chemicals. Many releases to be considered involve high-

momentum two-phase discharges. This may occur from emergency venting

of vessels due to fire exposure or runaway reaction, or from accidental

breaches in vessels, pipes or sealing. In the case of storage of pressurized

liquefied gas, venting induces flashing of the superheated liquid; in the case

of thermal runaway reactions, gassy reaction products can be released with

the liquid phase. Releases involving two-phase flow exhibit specific

characteristics which can significantly influence the dispersion process. The

rather recent realization of the importance of these effects has led to the

2


3/36

development of two-phase dispersion models.A comparative review of these

models is proposed here. To clarify

the situation under study and to introduce basic definitions, a description of

the phenomena taking place in the dispersion process is first recalled.[ P.

Bricard et al., 1998]

Mathematical modeling of the dispersion of the vapors resulting from accidental releases of

volatile liquids and heavy gases has been helpful in establishing safe designs and operating

procedures for the transport and storage of flammable and toxic materials.

Increased recognition that most emergency release from chemical storage or process vessels

involve the escape of volatile, pressurized liquids has led to a need for predictive models of two-

phase liquid-vapor forced plumes (jets).In many postulated releases from chemical and process

plants, the material contained within the plant is ejected through a break as a two-phase jet or

cloud which then disperses into the surrounding atmosphere. The growing realization that most

emergency releases involve high momentum two-phase discharges even in cases of controlled

releases, has led to increased interest in modeling such jet releases. Releases of liquids from

pressurized vessels have high momentum. A rapid depressurization and expansion occurs just

outside the release orifice. During depressurization the liquid jet material is converted into

aerosol as a result of flashing to vapor. Following external expansion to atmospheric pressure,

turbulent mixing of the jet with the atmosphere occurs and the surrounding air and water vapor

are entrained into the jet at a rate proportional to the mean jet velocity. [M. Epstein et al.,1990]

The temperature of the jet decreases with distance from the release point, owing to the

evaporation of the jet aerosol by the entrained, relatively warm air. As a result the water vapor in

the entrained air may condense to a fog. The observed persistence of visible aerosol plumesproduced by the release of volatile liquids is probably due to condensation of the atmospheric

water content. Beyond some not very significant distance from the release point (source) both

liquid phases, jet material and water, are consumed by evaporation and the forced plume

behavior is one of an all-gas heavier-than-air release. As dilution continues and the jet slows

down, it begins to spread over the ground under the influence of gravity and loses its roughly

3


4/36

circular shape. As demonstrated later, in this intermediate jet slumping (or gravity current) phase

the rate of entrainment of air is still proportional to the mean jet flow velocity, with the

proportionality factor taken to be a constant. In the final phase of jet behavior, far from the

release point, the atmosphere eddies determine jet dilution behavior.

A theory for the spreading of a steady-state initially high-momentum two-phase jet in a

horizontal wind is proposed here, in which it is assumed that for some rather large distance

downwind of the release location (in some cases of the order of thousands of meters), the effects

of buoyancy and/or atmospheric turbulence on the inflow or entrainment velocity may be

ignored in comparison with the effects of turbulence generated by the jet. This assumption is

what distinguishes the present work from most of the previous modeling efforts, which are based

on the notion that the rate of engulfment of air by the jet is determined by the atmospheric

boundary layer and/or by gravity beginning at relatively small distances from the source. In the

present model only two distinct phases of jet entrainment behavior are considered: mixing by the

jets self-generated turbulence in the near field and mixing by environmental turbulence in the

far field. Moreover a fairly sharp transition point at which near-field entrainment behavior

changes to far-field behavior is assumed. The intermediate phase involving the inhibiting effect

of buoyancy on mixing into the jet is regarded as insignificant.

1.1 Factors affecting two-phase toxic releases

The main factors affecting the consequences of a two-phase release are:

1) The scale and duration of the release, and the rate at which material becomes airborne,i.e. the source term

2) The rate at which the material disperses in the atmosphere and the direction of the wind,i.e. dispersion;

3) The toxicity of the material;

4) The population distribution surrounding the hazardous plant and their response to theemergency. . [Fauske et al.,1988]

4


5/36

.

Fig 1.1:The path of a two-phase release into the atmosphere.

1.2 The three regions of two-phase jet dispersion model

1.2.1 Depressurization Zone

Here aerosol formation takes place. In this region the jet undergoes a rapid

depressurization to or slightly below ambient pressure within a distance from the release

orifice of approximately two orifice diameters. It is within this zone that the liquid

disintegrates into fine droplets in the lo-50pm size range. The jet expansion in this region

is rapid and due almost entirely to vapor production; its interaction with the atmosphere is

negligible and very little mixing between jet and ambient occurs. The depressurization

zone is treated with a standard liquid flash and momentum balance equation, as discussed

by Fauske and Epstein. The trajectory and dilution of the jet in the intermediate, elevated

jet zone is predicted with a two-phase version of the buoyant jet entrainment models that

were originally developed for smoke stack or cooling water discharge plumes

1.2.2)Jet trajectory zone:

This is two dimensional zones for above ground level releases. This zone begins

when the jet is depressurized to atmospheric pressure. The forced plume (jet) is not

necessarily released at ground level and the component of initial jet velocity, u,, is not

necessarily parallel to the horizontal ambient flow (wind), but may be inclined away from

it, horizontally. Thus near the rupture opening (or vent) the solution of the equations of

motion may require that the jet rise. However, further downstream the heavier-than-air jet

5


6/36


7/36

valve. Since safety valves or rupture disks can be designed to direct jets upward, the

consequence of the release is far less serious than would be the case if the storage vessel itself

ruptured, giving rise to a liquid release near ground level. After a sufficient amount of air is

entrained, the jet release will be carried along with the ambient wind. If, however, the release is

significantly heavier than air, the jet could fall to the ground before this passive state is reached.

[Woodward J.L.,1989]

Fig. 1.2: Flow regimes of a high-momentum two phase release and coordinate system.[ M. Epstein et al., 1990]

7


8/36

Fig. 1.3: Idealized cloud section: side and cross-sectional views of a two-phase, elevated

continuous release at various stages of dispersion [Brown R. et al., 1962].

1.3 Chemical plume

Fig. 1.4: A chemical plume emitted from finite source decreases in concentration aschemical carried away from the source and dispersed by air and differentconcentration levels.

In the case of two-phase toxic releases, the source terms of prime interest to HSE arise from loss

of containment accidents in plant processing or storing superheated (i.e. pressurized so that the

8


9/36

liquid is above its boiling point at atmospheric pressure) liquefied gases. The transfer processes

associated with the subsequent discharge of material to the atmosphere are complex, and involve

non-equilibrium two-phase flashing flow. rapid expansion of the jet and further flashing of

liquid as the jet emerges from the breach, break-up of the liquid into droplets and aerosol with

possible liquid rain-out, and entrainment of air into the jet as it expands. Air entrainment

rapidly lowers the partial pressure of the vapor, thus causing further evaporation of the liquid

droplets, with much of the latent heat of vaporization coming from the sensible heat in the air as

it is cooled to the liquid droplet temperature although some will be taken from the liquid also).

These processes have been described comprehensively [Fauske et al.,1988].

1.4 Dispersion Process

The release is supposed to originate from a relatively small hole so that continuous, i.e.

quasi-steady, conditions at the outlet can be assumed. The cloud is defined as the smallest

control volume containing the contaminant. In its first stage, where its initial momentum

dominates, the cloud will also be referred to as jet. In most cases involving two-phase releases,

the flow is choked at the exit and an external depressurization zone, where the pressure decreases

down to the atmospheric pressure, is formed. When the exiting liquid is sufficiently superheated

with

respect to ambient conditions, it is atomized by violent vaporization .Otherwise, the liquid or

two-phase mixture is disintegrated due to liquid surface instabilities aerodynamic atomization..

Downstream from this region, air entrainment at the perimeter of the cloud becomes important,

which causes it to further widen .At least for some distance; the cloud may be dense, i.e. heavier

than air, as a result of high molecular weight(e.g. chlorine).or low temperature and airborne

droplets (e.g. evaporating ammonia).

The dispersion of the contaminant in the atmosphere can be described in terms of cloud

trajectory and dilution. From an integral point of view, the trajectory is given by a momentum

balance on the cloud; the main effects involved are cross-wind, gravity and friction on the

ground after touchdown. The dilution is controlled by the rate of air entrained in the cloud. Near

the outlet, this is governed by the turbulence generated by the jet itself; it is then controlled by

atmospheric turbulence when the jet velocity has decreased close to that of the ambient wind.

9


10/36

The dilution is controlled by the rate of air entrained in the cloud. Near the outlet, this is

governed by the turbulence generated by the jet itself; it is then controlled by atmospheric

turbulence when the jet velocity has decreased close to that of the ambient wind. Moreover, the

interaction with a cross-wind induces an enhancement of the entrainment rate. In the case of

dense clouds, gravity may also have an effect on air entrainment, related to gravity-induced

turbulence as well as suppression of atmospheric turbulence due to stable stratification. In the

following, the region of passive dispersion due to atmospheric turbulence only is referred to as

the far-field and the upstream region as the near-field.

The dispersion process may be significantly affected by the presence of an aerosol phase. First,

two-phase releases can lead to much higher discharge mass flow rates than single-phase gas

releases and, thus, increase the hazard zone distance. Moreover, the jet density may be

significantly higher. It can firstly be increased by the mere presence of the liquid phase.

However, this is only significant very close to the outlet, where the liquid mass fraction averaged

over the jet cross-section is not negligibly small. The aerosol effect on jet density is mainly due

to phase change phenomena. When the liquid contaminant evaporates, the jet may significantly

cool down and, thus, increase in density. A gas which has a smaller molecular weight than air

like ammonia can then behave as a heavy gas. The cooling process may also lead to the

condensation of the entrained humidity. If the contaminant is hygroscopic, this can lead to its

persistence to significantly larger distances from the outlet. The formation of the aqueous aerosol

will cause the mixture to warm up more rapidly and have less density. Furthermore, a part of the

liquid may not remain airborne in the jet and fall to the ground where an evaporating pool could

build up; such a pool may also be formed from the jet impingement on a surface. This so-called

rainout could induce a drastic reduction of the downstream contaminant concentration but

increases the danger close to the source as well as the duration of the dispersion [P. Bricard et

al., 1998].

CHAPTER 2

TWO PHASE CHEMICAL PLUME MODEL

10


11/36

The dispersion models which take into account the presence of an aerosol phase have

appeared only recentlyliteratures. They are either integral or multi-dimensional models.Integral

models are obtained by integrating the balance equations for mass, momentum, energy and

species over the cloud cross-section.Integral models are obtained by integrating the balance

equations for mass, momentum, energy and species over the cloud cross-section.

A schematic representation of the high-momentum two-phase plume is shown in Figure 2.1The

jet is inclined and released above ground level. The jet passes through several physical regimes

along its path. The regimes are modeled separately in the following sections.

Fig. 2.1: Two Phase Jet Release[Bricard P.,1998].

2.1 Depressurization zone

The jet expansion in this region is rapid and due almost entirely to vapor production; its

interaction with the atmosphere is negligible and very little mixing between jet and ambient

occurs. Thus friction, heat and momentum exchange with the ambient environment at pressure P,

are assumed to be negligible.

In an emergency release situation, liquid at high pressure usually escapes from a vessel whose

stagnation condition is sub cooled and exit choked mass flux well-represented by,

11


12/36

G= CD2P0-Pj,s(T0)j,f (1)

where PO is the stagnation pressure; Pj,s(To) is the vapor (or saturation) pressure of the escaping

jet material corresponding to the liquid stagnation temperature T0; and j,f is the density of the jet

materials liquid phase. This Bernoulli type expression shows reasonable agreement in

experimental comparisons. Equation (1) is derived below:

let, P1= P0 ;

P2= Pj,s(T0)

form of Bernoullis equation,

P1+u122=u222+P2

P1-P2=u222 As u1 =0;

P1-P2=u2222

We know,

G2=Kgm2sec: and also , u22= Kgm2sec

We can write, u22= G;

P1-P2=G22

G2=2P1-P2

12


13/36

G=2P1-P2

G=CD2P1-P2

Thus equation (1) is derived,

A space wise calculation of two-dimensional (or ax symmetric) jet pressure and velocity profiles

is difficult. However, it is possible to estimate the fully expanded jet properties at the end of the

depressurization zone, when the jet pressure everywhere has approached the ambient pressure.High pressure flashing liquids produce extremely fine droplets, so that phase velocities and

temperatures are quickly equalized. The principle of conservation of momentum flux applied to

this homogeneous mixture provides a simple relation between the fully expanded jet velocity u,

at\ the end of the depressurization zone and the break (or choke) plane quantities:

ue=Gj,f+Pj,sT0-PG (2)

To obtain the quality of the fully expanded jet, x,, we invoke the condition of constant

stagnation enthalpy.

ho =hj,f,sP+xehj,f,g(P) (3)

The kinetic energy of the two-phase jet is notably quite low (relative to that of an all-gas sonic

discharge) and therefore is not included in the above equation. For the all-liquid discharge

considered here, Equation (3) becomes,

xc= cj,f[T0-Tj,sPhj,fg (4)

The area of the depressurized jetA, is related to the area of the breakAbthrough the equation for

the conservation of mass,

13


14/36

Aa=Abubbuaa=AbGucc

in which the expanded jet density,e, is given by the definition,

e=xej,g,sP+1-xej,f,sP-1

(5)

The volume occupied by the liquid at the end of the depressurization zone can readily be shown

to be negligible. Thus the equilibrium quantity j,g,sP can be identified with the mass of the

vapor per unit spatial volume, i.e.

j,g,e=j,g,sP (6)

The mass of the liquid phase with respect to the same spatial volume is given by

j,g,e=1-xec (7)

The quantities ue,Ae,j,g,e and j,f.e are used as input to the jet trajectory and dilution model.

[ M. Epstein et al., 1990]

2.2 Air born jet trajectory zone

This zone begins when the jet is depressurized to atmospheric pressure. The forced plume(jet) is not necessarily released at ground level and the component of initial jet velocity, ue is not

necessarily parallel to the horizontal ambient flow (wind), but may be inclined away from it,

horizontally. Thus near the rupture opening (or vent) the solution of the equations of motion may

require that the jet rise. However, further downstream the heavier-than-air jet is influenced by the

14


15/36

gravitational field and its trajectory is similar in shape to that of a projectile. The trajectory

zone or, equivalently, the flight of the jet terminates upon contact with the ground.

The equations are written to take account of a steady, horizontal, wind. The wind direction or,

velocity, Wdoes notchange with altitude z, the jet lying in the x, z plane (see Figure 1.2).The

mean velocity inside the jet is denoted by the symbol u. The centerline of the jet makes an angle

with the horizontal, s is the distance along the jet, A is the cross-sectional area of the jet,

and C is the perimeter of the cross-section. For simplicity, we assume that inside the jet the

velocity, vapor and liquid concentrations, and temperature are uniform in the crosswind plane.

Over the distance the jet is airborne its cross-section is taken to be circular and its radius Ris

assumed to be small compared with the centerline radius of curvature (slender plume

approximation).

Once the jet is depressurized to atmospheric pressure, the jet expands solely by entrainment of

ambient air. The principal assumption of this theory is that, for some significant distance from

the source, the effects of atmospheric turbulence may be ignored, provided the initial momentum

of the jet is high enough. Some performed an analysis on a limited set of experimental data on

buoyant jet flow discharged into a uniform cross stream to determine the jet entrainment velocity

due to the jets self-generated turbulence. For jets with a two dimensional trajectory, then

expression becomes,

ven=1/2E0u-Wcos+E1Wsin

(8)

Where the term u-Wcos represents the relative velocity of the jet with respect to the wind.

Actually, Equation (8) is a combination of the entrainment model proposed by Hirst and the

entrainment model of Ricou and Spalding for vastly different jet and ambient densities, p and

pm, respectively. Precise definitions of and are given below. Equation (8) reduces to the

entrainment function of Hirst in the limit .The constants of proportionality Es and Et are

called the entrainment coefficients. Some authors have verified through measurements that for

pure momentum jets (E1 = 0) values of E0 lie within the range 0.06 to 0.12.

15


16/36

If a uniform velocity profile is used E0 = 0.12 results in the best agreement between theory and

experiment. From data on buoyant water jets discharged at varying angles into flowing aqueous

salt solutions, the specified the values E0 = 0.057 and El = 0.513. In the calculations which

follow, we shall adopt Equation (8) and use values of E0 and El of 0.1 and 0.2, 0.5, respectively.

The jet trajectory can now be determined from the conservation equations for mass, momentum

and energy, using the entrainment equation to determine the rate of mass, momentum and energy

addition into the jet.[ M. Epstein et al., 1990]

The equation for the conservation of mass of the discharge material,

ddsj,g+j,fuA=0 (9)

The equation for the conservation of mass of the air,

ddsauA=a,venC (10)

The equation for the conservation of mass of the atmospheric water content,

ddsw,g+w,fuA=w,g,venC (11)

Note that in addition to the liquid aerosol of the discharge jet material we allow for the

possibility of water droplet formation via condensation of the entrained water vapor within the

cold jet (see Equation (11)). Neglecting the volume occupied by the jet aerosol and water

droplets, we can express the overall spatial density, p, of the three components two-phase

Mixture as the sum of the spatial densities of the components,

=j,g+j,f+a+w,g+w,f (12)

16


17/36

The momentum equation in the vertical direction is a balance between plume momentum and

mass,

ddsAu2sin=-g-A (13)

where is density of the air/water vapor atmosphere, namely

=a,+w,g, (14)

The momentum equation in the horizontal direction is a balance between jet momentum and

momentum entrained,

ddsAu2cos=WddsAu (15)

Conservation of heat may be written in the following form,

ddsuAh=a,venCha,+w,g,venChw,g,

(16)

where h is the overall enthalpy of the mixture within the jet; it is given by the definition

h=j,ghj,g+j,fhj,f+aha+w,ghw,g+w,fhw,f (17)

The energy equation, Equation (16), may be readily converted from enthalpy to temperature as

the dependent variable by introducing the thermodynamic relation h(T) for each species,

assuming the vapor phases are ideal and the liquid phases are incompressible. The height z of the

jet above ground level is, in differential form,

dz ds=sin (18)

and the horizontal distance x from the rupture opening is given by,

17


18/36

dxds=cos (19)

Note that in writing the above equations, we have neglected the length of the depressurization

zone. This length is typically quite small relative to the range of the jet. To predict the densities

of the vapor phases in the presence of their condensed phases it is assumed that thermodynamic

equilibrium is locally achieved within the jet. Thus, utilizing the ideal gas equation of state, we

have,

Ideal gas law is,

PV=nRT

From this we get,

j,g=Pj,sTRjT (20)

w,g=Pw,s(T)RwT

(21)

The pressure within the jet is equal to the atmospheric pressure P. This requirement gives,

P=j,gRjT+w,gRwT+aRaT (22)

Finally, since the cross-section of the jet is taken to be circular, the radius R, circumference and

cross sectional area are related by the expressions,

C=2R=2(A)1/2 (23)

2.3 Boundary conditions

The above equations express the mathematical formulation of the model and are sufficient to

determine the nine coupled unknown quantities, j,g , w,f,a,u ,A ,T and as a function of

the

Coordinates x, z. These quantities are subject to the following initial conditions when x = 0 and

18


19/36

z = z0,

j,g=j,g,e ,j,f=j,f,e (24)

w,g=w,f=a=0 (25)

u=ue , A=Ae (26)

T=Tj,s ,=0 (27)

The initial thermodynamic state of the mixture, at the end of the depressurization zone, is one of

jet material aerosol in equilibrium with its vapor phase. Downwind of the release point humid airis entrained into the jet causing further evaporation of the liquid jet material. At first water exists

within the jet only as vapor and Equation (11) is integrated with w,f=0 .[ M. Epstein et al.,

1990]

As a result of a combination of the cooling of the jet by the evaporation of the released liquid jet

material and the buildup in the water vapor concentration by entrainment of the surrounding air,

the water vapor may become saturated so that both vapor and liquid water are present in the

system. In some systems there is a tendency of jet material vapor to be soluble in liquid water.

This effect is ignored in the present model. As the amount of air increases, the liquid water also

begins to evaporate. Ultimately all the water and all the jet material are in vapor form. Usually

the water is the last to evaporate. The evaporation of the water in the far field will tend to

prolong the persistence of the cool jet beyond the distance to ground contact.

2.4 Ground- level dispersion zone

This region begins at touchdown or, in mathematical terms, when the elevation of the

centerline of the jet is equal to its radius (i.e. z = R, see Figure 1.2). The atmospheric dispersion

of the ground-level jet is affected by both entrainment of the atmosphere and gravity spreading.

19


20/36

Gravity spreading is driven primarily by the excess hydrostatic pressure caused by the density

difference between the jet and the atmosphere.

In the initial phase of ground-level jet behavior, the turbulent motion within the jet is normally

still due to the movement of the jet itself and, accordingly, Equation (8) forven remains valid.

Of course in the final phase of jet behavior, far downwind of the source, the energy containing

eddies of environmental turbulence dominate mixing and the ground-level jet growth is strictly

due to the action of the wind. The far-field jet growth is assumed to be well-described by the

Gaussian plume, point source solution to the atmospheric advection-diffusion equation.

Accordingly,\ a realistic equation for the entrainment velocity in the far field is,

ven= WCddxyxz(x) (28)

Which when combined with the conservation of mass equation gives the point source Gaussian

solution for the ground-level concentration of the released species in the far field of the jet. In the

above equation, yx andz(x) are the dispersion coefficients (or standard deviations) for the

crosswind plane. The most widely used y and z functions are the so-called Pasquill- Gifford

curves. These empirically determined functions depend on the atmospheric stability classes. The

field observations reported by Burgess et al. suggest that the dispersion of heavier-than-air

ground-hugging vapors is best represented by Gaussian theory when y and z are based on the

most stable atmospheric condition, stability class F, regardless of the actual state of the

atmosphere. This recommendation is followed here.

Close to the source, the entrainment velocity given by Equation (8) greatly exceeds that given by

Equation (28) and vice versa at distances far from the source. One would expect, therefore, a

fairly definite transition point (or perhaps a short transition zone) at which the character of the jet

changes when the near- and far-field entrainment velocities are comparable. Therefore, during

the course of a calculation a switch is made from Equation (8) to Equation (28) at the location in

which both equations predict the same entrainment velocity, thereby merging the high-

momentum jet theory with the Gaussian plume model(A computer model used to calculate air

pollution concentrations. The model assumes that a pollutant plume is carried downwind from its

emission source by a mean wind and that concentrations in the plume can be approximated by

20


21/36

assuming that the highest concentrations occur on the horizontal and vertical midlines of the

plume, with the distribution about these mid-lines characterized by Gaussian- or bell-shaped

concentration profiles). [Nikmo J. et al., 1994]

It is obvious that by equating the two entrainment velocity laws, Equations (8) and (28), the

intermediate phase of jet behavior involving buoyancy-dominated dispersion effects has been

assumed to be insignificant. This assumption implies the dominancy of inertial force over the

buoyancy force. The ratio of jet buoyancy to inertia is given by the Richardson number, defined

by,

Ri=2g(-)zu2

(29)

The work of Ellison and Turner showed that the entrainment is not buoyancy dominated if

Ri0.1 As we shall see later on, this criterion is indeed satisfied for volatile liquid chemical

releases of practical interest. It is worth noting that observations of smoke plumes from a large

power station suggest the sudden transition assumed here, from jet momentum drive entrainment

to wind dispersion.

In the present model, the ground-level dispersion is handled in the same way as the elevateddispersion, except that the momentum equations, Equations (13) and (15), and kinematic

equations, Equations (18) and (19), are replaced by corresponding equations that account for

gravity compaction and sideways spreading. Assume that the cross-section of the ground-level

jet is rectangular, with height 22 and radius (half-width) R. Note that z still represents the

Vertical distance from the ground to the centerline of the jet. Beyond ground contact, Equation

(23) for the perimeter of the jet is replaced by the expression,

C=4z+A/(2z) (30)

This relation for the rectangular jet is derived by not allowing entrainment to occur at the jet-

ground interface. The half-width of the jet R at ground contact is calculated by setting the

perimeter and area of the rectangular ground-level jet equal to those of the circular elevated jet.

21


22/36

This ensures that in the post ground contact region the masses within the jet are conserved and

that the continuity and smoothness of concentrations- and temperature-with-distance are

preserved. A discontinuity in R will exist at ground contact.[ Heinrich M. et al.,1988]

The momentum equation applied to the side edges of the jet is,

ddsAuv=g(-) (31)

Where u is the velocity of the jet in the lateral direction (or the spreading rate). The momentum

equation in the axial direction is,

ddsAu2=WddsAu-gdds[-Az] (32)

After jet touchdown, these equations are substituted for Equations (13) and (15). The motion of

the material interface represented by a side edge of the jet is given by the kinematic condition,

udRds=ven+v

(33)

where ven and v are the velocities normal to the edge. With the aid of the equation for the cross-

sectional area of the plume, namely A = 4zR, Equation (33) can be written as follows,

dzds=zAdAds-4z2Aven+vu

(34)

which replaces Equation (18) after plume touchdown. Finally, during the gravity spread

calculation, Equation (19) takes the form,

dx ds=1 (35)

22


23/36

( i.e. = 0 )

It should be mentioned that the only new variable that appears after the transition is made from

the airborne plume to the ground-level dispersion is the spreading velocity v; it is assigned the

initial condition v = 0 at the location of ground contact.

From this system of equations we can compute the jet temperature, released species and

entrained water concentrations, jet boundaries, jet velocity, etc. as a function of distance from the

source. To utilize available computer subroutines, the equations were converted to an equivalent

coupled system of first order ordinary differential equations by expanding the derivatives in

Equations (9)-(11), (13), (15), (16), (31), and (32).

2.5 Integral Models

Integral models can either deal with both the near-field and far-field or consider the near-

field only. In the latter case, they are regarded as source term for a far-field dispersion model.

The models are all based on the balance equations for mass, momentum, energy and species,

integrated over the jet cross-section; they mainly differ by the adopted closure relations. A

common frame based on the mixture balance equations is first proposed to enable a comparison

between the models. Then, the closure relations, or sub models, are described.[Bricard P. et

al.,1998]

2.5.1 Balance Equations

In the following, we assume that the cloud (or jet) centre-line remains in the wind

gravity plane. Letsbe the curvilinear coordinate of the cloud centre line, (x, y, z) its Cartesian

coordinates such that wind and gravity are oriented along the x and z directions, respectively

(Fig.4) and the angle between the jet centre-line and the horizontal axis such that,

dxds=cos

23


24/36

dzds=sin

With A being the cloud cross-section area, , u and h the mixture density, velocity and

specific enthalpy, respectively, and c the contaminant mass concentration (in kg/m3) in the cloud,

the balance equations for continuous (steady-state) release can then be written as,

Mixture mass balance is ,

ddsAudA=E (36)

Contaminant mass balance is,

ddsAcudA=0 (37)

Mixture enthalpy balance is,

ddsAuhdA=Eha (38)

Mixture x- momentum balance is,

ddsAu2cosdA=Eua+Fx (39)

Mixture y- momentum balance is,

ddsAu2sin=Fz (40)

Where E is the so-called entrainment function representing the rate of air entering the cloud;

ua and ha are the velocity and the specific enthalpy of ambient air, respectively. F x and Fy are the

forces which act on jet in horizontal and vertical directions respectively [Bricard et al.,1998].

24


25/36

2.6 Forces on the cloud

Models which do not include wind and gravity effects on the jet trajectory assume

constant jet momentum so that Fx = Fy=0. Other models contain a gravity term as a contribution

of Fz .Most authors also introduce an additional drag force FD due to cross wind. This force is

supposed to be act in the direction perpendicular to cloud axis. Its component in x and y

directions are given as;

For x- direction,

FDx=CDC(0.5)a(uasin)2sin

For y direction,

FDz=CDC(0.5)a(uasin)2cos

where CDis a drag coefficient. A constant value of 0.3 is adopted mostly. The addition of a drag

force may be useful to take into account the pressure gradientperpendicular to the jet flow;

however, the drag coefficient should be drastically lower than for the flow around a solid

cylinder due to suction effects [ Bricard P. et al.,1998].

25


26/36

CHAPTER 3

SOME TWO PHASE MULTI DIMENSIONAL MODELS

We will discuss some two phase multidimensional models that we consider subsequently

are the one of Wiirts et al., (1996), Vandroux-Koenig et al.,(1997) and Pereira and Chen (1996)

3.1 Wiirtz et al.,Model

This three-dimensional model includes the mixture mass, momentum and energy Balance

equations as well as the mass balance equation for the contaminant component. The mixture is

composed of the contaminant (liquid and/or gas) and the ambient gas (no humidity condensation

is considered.Thermo dynamical equilibrium is assumed, i.e. all the components share locally

the same temperature and pressure. The two-phase mixture is supposed to behave ideally i.e.

Raoults law is used for the calculation of the partial pressures.A single-phase turbulence model,

based on the eddy diffusivity concept, is adopted and modified to take into account anisotropy

effects. The working fluid is treated as a mixture of two fluids in thermodynamic equilibrium.

Both fluids may contain several components. The fluid kinetics is described by assuming a slip

velocity between the liquid phase and the ambient gas.A vertical slip velocity is allowed for

between the liquid and gas phase. Special attention was paid on the models ability to handle

complex terrain. In particular, liquid deposition on solid surfaces is taken into account. Remarks

already made regarding the assumption of thermodynamic equilibrium in the integral models

26


27/36

also apply here. The first validation tests performed by the authors have shown the better

performance of this model against a 1-D model when obstacles are present.

3.2 Vandroux-Koenig et al., Model

This model is devoted to the prediction of the near-field dispersion of liquefied propane.

It is a EulerianEulerian two-fluid model which considers three components: propane (liquid and

vapor)., dry air and water (liquid and vapor). The condensed water is included in the gas phase to

form a homogeneous fog mixture. Balance equations are written for the mass of each constituent,

for the momentum of the gas mixture and of the propane droplets, and for the gas mixture

energy. The temperature of the propane droplets is supposed to be uniform, equal to the

saturation temperature at atmospheric pressure, so that no energy balance equation is needed for

the liquid phase. To close the system of equations, a single-phase turbulence model based on the

Prandtls mixing length theory is introduced for the gas phase. For interfacial transfers, a

constant droplet diameter is assumed throughout the calculation. The momentum and heat

interfacial transfer terms are calculated from a drag and a heat transfer coefficient, respectively,

for rigid spheres. The mass transfer is modeled by assuming that the heat transferred from the

gas phase to the droplets completely contributes to vapor production.

According to the authors, the propane droplets are predicted to persist much further downstream

than experimentally observed. Several possible causes were investigated for this underestimation

of droplet evaporation. Neither the assumption of a constant diameter nor the fact that no initial

radial velocity was taken into account could explain this under evaluation.On the other hand, the

assumption of a uniform temperature in the droplet or the inadequacies of the turbulence model

(the droplets may in reality disperse more than the gas phase)have been proposed as possible

explanations. More parametric studies as well as the comparison of the prediction with the

results of controlled laboratory experiments adapted to the validation of the uncertain aspects of

the model are required to solve these inconsistencies.

3.3 Pereira and Chens Model

This Eulerian-Lagrangian model is also devoted to the near-field dispersion of liquefied

propane; the single-phase version developed for the far-field dispersion is not considered here.It

is constituted of a Eulerian description of the mixture phase composed of air and propane vapor

27


28/36

(no humidity is taken into account) having the having the same velocity and temperature but

different volume fractions, and a Lagrangian modeling of the droplet phase which is composed

of various droplet-size groups having their own initial characteristics (velocity, temperature,

diameter,). The effect of the dispersed phase on the gas phase is limited to the mass,

momentum and enthalpy transport due to the phasechange process. For each of the two size

classes of droplets, the equation of motion is written. The interfacial force is determined by using

a drag coefficient valid for solid spheres and with the relative velocity evaluated from the mean

local gas velocity.

The Lagrangian approach has the advantage to account for the instantaneous flow properties

encountered by the particles. A second cited advantage of the EulerianLagrangian approach is

the possibility to readily handle the evolution of a distribution of particle diameters, which

remains difficult to predict in a EulerianEulerian scheme. However, transient situations are

more easily solved by a EulerianEulerian approach. This seems to favour the choice of

Lagrangian models for the description of the continuous releases considered here. The first

advantage is, however, not used by Pereira and Chen since only the mean local gas

characteristics are taken into account in the droplet equations.

28


29/36

CHAPTER 4

EXPERIMENTAL DATA AND MODEL VALIDATION

The tests whose results we have used for model comparisons are: Desert Tortoise (DT) 4

liquid ammonia (NH,) release; Goldfish (GF) 1 and 3 liquid anhydrous hydrofluoric acid (HF)

release; and Energy Analysts (EA) 1 and 2 small-scale liquid ammonia and propane releases.

The choice of these tests is based on the availability of good experimental data. The test

parameters are shown in Table 1. Additional test details can be found in the references cited

below. It should be noted from the table that the liquids were released in the direction of the

wind approximately 1 m above ground level as a horizontal jet (=0). The only exception was

EA ammonia test 1 where the release angle from the horizontal 0=100.Thus in most of the

tests the jet, made contact with the ground a short distance downwind from the source. The

release heights were not reported for the Goldfish series of tests. For the purpose of carrying out

HF dispersion calculations, a release height z0=0.79m was assumed. Ammonia was released

from this height during the Desert Tortoise series of tests, which were conducted at the same test

site as the Goldfish tests.

The Desert Tortoise series test results were reported By Goldwire et al. and Goldwire. In

Figure 5, the predicted profile for the DT4 ammonia plume is presented out to 300 m from the

source. Water droplet formation is predicted to occur essentially at the source. The jet is

predicted to impact the ground at 2.92 m downwind from the source. Goldwire et al. reported

that a noticeable pool of liquid ammonia persisted within at least 3 m of the source. The liquidammonia and water aerosols are predicted to evaporate completely at 40.7 and 201 m downwind,

respectively. The actual plume was visible to a point approximately 400 m downwind,

indicating, perhaps an over prediction of the entrainment rate. The calculation suggests that the

atmospheric water content is responsible for long range visibility of the plume and not the liquid

ammonia.[ Goldwire et al.,1986]

29


30/36

The temperature and ammonia concentrations versus distance from the source are compared in

Figures 4.4 and 4.5. The predicted quantities represent average values, while the experimental

data for the temperature in the far field and the concentration are the measured values close to

the centre of the jet at an elevation of 1 m. The temperature measurements in the near field, at the

3, 6, and 9 m locations, represent those of the liquid ammonia deposited on the ground just

downwind of the source. As seen in Figure 4.4, the model is in excellent agreement with the

measured temperatures in the near and far field. The model performance in the far field,

however, may not be as average temperatures while the measurements correspond to jet

centerline values at an elevation of 1 m. In Figure 5, the model is seen to predict the ammonia

concentration at the 100 m position and under predict the concentrations at the 800 and 1000 m

positions by factors of 2.0 and 3.0, respectively (see also Table 2). This level of accuracy is

considered to be reasonable in the field of modeling the dilution of chemical releases in the

atmospheres. Again, the model results are average values, while the experimental data are the

measured values at the centre of the jet.

Interestingly enough, the entrainment velocity v, given by Equation (8) did not fall below u,,

given by Equation (28) during the course of the calculation that produced the NH3 concentration

versus distance curve of Figure 5. Thus for a distance downwind of the source of at least 3000 m,

it does not appear to be necessary to invoke the presence of atmospheric turbulence due to wind

to explain the measured NH3 jet dilution. While at first sight this result may seem surprising, it

should be mentioned that many other examples have been reported in the literature in which self-

generated turbulence dominates the jet mixing process far downstream of the source.

30


31/36

Table 1: Experimental conditions for field tests

The data which show that Equation (8), with El = 0 and E0 = 0.1, can be applied to plumes

extending 3000 m above an oil fire . The distance range over which Equation (8) is valid was

extended by another order of magnitude: Wilson et al. and later on Sparks successfully applied

the formula to describe plumes of hot gas emitted from erupting volcanoes. Such plumes reach

heights greater than 40000 m. The observations of smoke plumes released from a power plant

stack into a cross-wind showed that the transition from plume-to-atmospheric generated

entrainment occurs at distances as large as 425 m from the chimney. The applied an entrainmentformula similar to Equation (8) to explain data collected on chimney plumes in a cross-wind and

found the formula to be valid over a distance of approximately 1800 m from the chimney. Note

that in such chimney plumes only buoyancy is responsible for plume momentum at the source;

yet self-generated turbulence dominates the entrainment process far downstream from the

chimney. Finally, with regard to the DT4 NH3 release simulation, it is significant to note that the

31

TEST Ref. liquid Tj,sP

(K)

P0

(MPa)

T0

(K)

T

(K)

RH

(%)

Ab

(m2)

z0

(m2)

0

W

ms-1

DT4 Goldwire

et al.,

NH3 240 1.4 297 306 21.0 0.007 0.79 0 4.5

GF1 Blewitt

et al.,

HF 293 0.87 313 310 5.1 0.0014 - 0 5.6

GF3 Blewitt

et al.,

HF 293 0.91 312 300 27.9 0.00046 - 0 5.4

EA1 Pfenninget al.,

NH3 240 0.83 292 292 49.0 0.00034 1.22 10 7.62

EA2 Pfenning

et al

C3H8 231 1.03 289 289 51.0 0.000105 0.8 0 5.08


32/36

predicted Richardson number is 0.02 at the source, rises to a maximum value of 0.11 at about

200 m downwind and then decays to zero in the very far field. Clearly, for high-momentum jets,

gravity is not an important feature of the entrainment process.

Distance

Downwind

(m)

Concentration (mole

fraction)

DT4

Concentration (mole

fraction)

GF3

Measure-

ment

prediction Measure-

men

prediction

100 0.06 0.065 - -

300 - - 0.0095 0.01

800 0.02 0.0092 - -

1000 - 0.0013 0.0015

3000 0.05 0.0016 0.00018 0.00029

Table 2: Comparison of theory with concentration measurements

The hydrogen fluoride dispersion experiments were conducted and reported by Blewitt et al.

The liquid HF and atmospheric water condensate for Test GFl are predicted to evaporate in the

near field, at approximately 24 and 29 m, respectively. During GFl the jet was visible to a

distance of approximately 700 m from the source. Thus the visibility of the jet in the far field

cannot be explained by the presence of water droplets alone. Hydrogen fluoride and water vapor

exhibit a strong chemical interaction which results in the formation of a condensed phase

(aerosol) of HF/H2O solution droplets. These relatively stable, chemically produced droplets

would be expected to persist out to significant distances from the source.

The present model is based entirely on consideration of local phase-change thermodynamic

equilibrium. Chemical reactions between the jet material and water vapor are not considered.

32


33/36

The measured centerline temperatures and HF concentrations for Tests GFl and GF3 were found

to agree well with the cross-wind averaged values of the model. The measured and predicted

temperature and concentration versus distance curves for Test GF3 are shown in Figures 4.4 and

4.5, respectively. The numerical solutions indicate that the application of an entrainment

equation for high momentum jets, namely Equation (8), is not valid for distances from the source

beyond about 80 m for the HF tests. At this point the transition is made between high-momentum

jet and Gaussian plume entrainment behavior. Neglecting entrainment associated with

atmospheric turbulence will lead to over-estimates of the concentration. The predicted maximum

Richardson number for both tests is only about 0.02 at about 10 m from the source, indicating

once again that buoyancy does not influence the entrainment process in an essential way. Note

from Table I that the break area and stagnation pressure for the NH 3 test are much larger than the

corresponding quantities for the HF tests. This readily explains why the NH3 jet in Test DT4 was

dominated by the momentum flux from the source at all the concentration sampler stations, while

the momentum flux for the weaker (Test GF3) HF jet vanished some distance upwind of the first

concentration sampler, and beyond this point spreading was dominated by atmospheric

turbulence. Small-scale field releases of pressurized liquid ammonia and propane were

performed by Pfenning et al. (the Energy Analysts (EA) series). Using slides and video tapes, the

average profile of the visible portion of the jet was obtained. No vapor concentration or

temperature measurements were made.

In Test EAl (see Table 1) the liquid ammonia was released at a 100 incline from the horizontal.

Figure 8 shows the predicted profile of the ammonia jet for Test EAl. The visible jet extended

approximately 45 m downwind, with a maximum height of 3.7 m. This is to be compared with

the prediction of 40.4 m, based on complete water droplet evaporation, and the predicted

maximum height of 3.9 m. The predicted length of the visible propane jet for Test EA2 is 17.9 m

and the predicted maximum jet height is 1.9 m. The visible length and maximum jet height

measured were 24 m and 2.1 m, respectively. In both the EAl and EA2 releases the jet material

aerosol is predicted to evaporate completely be ore contact with the ground. This is consistent

with the observations of that in the tests no liquid accumulated on the ground [Pfenning et

al.,1987].

33


34/36

Figure 4.1 : Predicted jet profile for Desert Tortoise ammonia spill Test 4 (DT4)

Water fog formation At x=0.056m, height

Touchdown At x= 2.92m, height

Jet material liquid evaporated At x= 40.73m, height

Water fog evaporated At x= 201m , height

Figure 4.2: Temperature as a function of distance from source for Desert Tortoise

ammonia Test 4 (DT4).

Figure 4.3 : Ammonia concentration as a function of distance from source for Desert

Tortoise spill Test 4 (DT4)

Figure 4.4: Temperature as a function of distance from source for Goldfish hydrofluoric

acid spill Test 3 (GF3)

Figure 4.5: Hydrofluoric acid concentration as a function of distance from the source for

Goldfish hydrofluoric acid spill Test 3 (GF3).

34


35/36

Figure 4.6: Predicted jet profile for Energy Analyses ammonia release test (EAl)

CHAPTER 5

CONCLUSION

In this paper we have provided a modeling approach to determining the dispersion hazards

arising from the release of pressurized liquid chemicals. The dilution of the two-phase jet

produced by the release has been modeled taking into account the three regimes of jet behavior.

A one-dimensional source model describing the depressurization of the jet and two turbulent

entrainment models describing the elevated and ground-level behavior have been presented.

35


36/36

This review has been focused on the comparative description of two-phase jet dispersion models,

with special attention to elevated and continuous conditions. A description of available data for

two-phase jet dispersion was also given. Both integral and multidimensional models were

considered.

Most often, the integral approach has been adopted because of its good compromise between

model detail and computing timer cost, which is well adapted to hazard assessment. In integral

models, entrainment modeling remains an important issue, as reflected by the differences

between adopted formulations. Regarding two-phase flow phenomena, the homogeneous

equilibrium approximation is often used. The range where this is justified has, however, not been

defined in the case of high momentum jets. When this approximation is, at least partially relaxed,

the prediction of the initial drop size, drop trajectory and possibly evaporation is required. The

corresponding sub models show however a rather high degree of simplification and uncertainty.

Some propositions for their verification and improvement have been given. They include the

comparison of the sub models prediction to recent experimental data, the extension of models to

non-flashing two-phase releases and a rational criterion for droplet suspension in a jet.

Generally speaking, the models are insufficiently validated. To be able to give a firm judgment

of their quality, an independent evaluation is clearly needed. For this purpose, additional detailed

two-phase jet experiments should be performed.

Documents

Model of dilution of forced liquid