ChemicalEngineeeringScience v56 p4725 4736

7/25/2019 ChemicalEngineeeringScience v56 p4725 4736

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Chemical Engineering Science 56 (2001) 47254736www.elsevier.com/locate/ces

Sedimentation of polydispersed particles froma turbulent plume

Silvana S. S. Cardoso , Mehran Zarrebini

Department of Chemical Engineering, University of Cambridge Pembroke Street, Cambridge CB2 3RA, UK

Received 30 June 2000; received in revised form 12 April 2001; accepted 17 April 2001

AbstractParticle-laden plumes are ubiquitous in many industrial environments, including cement manufacturing, metallurgical processes

and fossil-fuel-based power plants. Once a plume is emitted into the atmosphere, the size of the particles plays a crucial role ontheir sedimentation patterns and on the risk they pose to terrestrial ecosystems and to humans. It is therefore of great interest toconsider and compare the behaviour of particles of dierent sizes emitted from an industrial chimney. In this paper, we present amodel that describes the dynamics and deposition pattern from gravity currents generated by axisymmetric particle-laden plumescomposed of polydispersed particles. Predictions of the deposition patterns of particles of dierent sizes are successfully comparedwith new laboratory measurements. We also present experimental and theoretical predictions for the mean particle diameter in thedeposit at the oor as a function of radial position. We consider environments of both innite and nite lateral extent. Finally,we illustrate and discuss the application of the model to the emission of chromium particles from a commercial chromium plating

plant. ? 2001 Elsevier Science Ltd. All rights reserved.

Keywords: Particles; Plume; Fluid mechanics; Sedimentation; Pollution; Environment

1. Introduction

Air pollution caused by dust emissions is currently be-ing given great attention by industry and legislation (USEPA, 1997; World Bank, 1999; WHO, 1999). Particu-late matters are emitted from various industrial sources,including coal-red thermal power plants, roasters andsmelters for the production of various metals, petroleumreneries, uidised-bed catalytic cracking units, cement

and fertiliser plants. The eects of particulate emissionson human health and terrestrial ecosystems can be verysevere, more so when noxious chemicals have been ad-sorbed on the surface of these particles. Small particles,of diameter of 10m or less, remain suspended in theatmosphere for long periods of time. These particles,when inhaled, can pass through the natural protectivemechanism of the human respiratory system, causing

Corresponding author. Tel.: +44-1223-31863; fax: +44-1223-334796.

E-mail address: silvana [email protected](S. S. S. Cardoso).

irritation, asphyxia, central nervous system depression,gastric intestinal track irregularities and pulmonary -

brosis (Schwartz, 1993; Kane, 1994; US EPA, 1996). Inparticular, the smaller particles, of 2m or less, are re-sponsible for most of the excess mortality and morbid-ity associated with high levels of exposure to particulates(World Bank, 1999). Furthermore, deposition of partic-ulates on vegetation may result in the reduction of plantgrowth and yields. Particulate air pollution also plays animportant role on the corrosion, erosion and soiling ofany exposed surfaces, such as buildings and cars (WorldBank, 1999).

In view of the greater risk to health posed by smallerparticulates in comparison to the larger ones (WorldBank, 1999), it is important to determine and comparethe behaviour in the atmosphere of particles of dierentsizes emitted from an industrial chimney. In particular, itis important to predict the concentrations of each particlesize in the air in the vicinity of the source of pollution,as well as their deposition rate on the ground.

In a recent paper, Zarrebini and Cardoso (2000) ex-amined the dynamics of and the deposition pattern arising

0009-2509/01/$- see front matter? 2001 Elsevier Science Ltd. All rights reserved.PII: S 0 0 0 9 - 2 5 0 9 (0 1 ) 0 0 1 4 3 - 9


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4726 S. S. S. Cardoso, M. Zarrebini / Chemical Engineering Science 56 (2001) 47254736

Fig. 1. Schematic of the experimental apparatus.

from axisymmetric particle-laden plumes. A turbulent

plume was released into an environment of nite verti-cal extent (Fig. 1). Upon reaching the maximum height,the plume spreads out radially, forming a current. Thesediment is suspended in the current by turbulence, but

particles continuously settle out of the lower boundaryof the current. As the sediment settles out, it is drawn

back towards the plume by a net inow driven by theentrainment of ambient uid at the plume margins. As aresult, a fraction of the settling particles are re-entrainedinto the rising plume. It was found experimentally thatsuch interaction between the particles in the environmentand the continuing plume plays a central role on the dy-

namics of the plume and on the sedimentation patternson the surrounding oor. Zarrebini and Cardoso (2000)used these experimental observations to develop a theo-retical description of the concentration of particles in the

plume, in the gravity current and in the surrounding en-vironment, as well as the rate of settling on the oor. Thetheoretical predictions were successfully compared withdata from the laboratory experiments.

The work of Zarrebini and Cardoso (2000) addressedthe simplied case of uniformly sized particles. How-ever, in most industrial processes involving particle-ladenows, as the examples described above, a range of parti-cle sizes is present. Such polydispersivity of particle sizeswill have important eects on the patterns of sedimenta-tion from the gravity current and on the re-entrainmentof particles into the plume, and thereby on the dynamicsof the continuing plume. Therefore, the concentrations ofthe particles in the environment surrounding the plume,as well as density of the deposit on the oor, will dependon the particle size distribution.

In this paper, we present a model that describes thedynamics and deposition pattern of radially spreadinggravity currents generated by a particle-laden plume com-

posed of polydispersed particles. Predictions of the de-position patterns are successfully compared with new

laboratory measurements. We also present experimentaland theoretical predictions for the mean particle diameterin the deposit at the oor as a function of radial position.We consider environments of both nite and innite lat-eral extent. Finally, we illustrate and discuss the applica-tion of the model to the emission of chromium particles

from a commercial plating plant.

2. Experimental equipment and methods

2.1. Experimental procedure

The experiments were conducted in a tank of crosssection 75 cm 75 cm and height 75 cm. The tank waslled to a depth of 30 cm with NaCl aqueous solution,with a concentration of approximately 3% (w : w). The

accurate concentration of this solution was measured byrefractometry. A mixture of fresh water and particles wasstirred continuously in a bucket of volume 10 l. The par-ticle concentration in this suspension was 6:00 g=l. Thesuspension was pumped continuously through a nozzleof diameter 7 mm positioned at the centre of the base ofthe tank (Fig. 1). The ow rate into the tank was con-trolled by a ow diverter; the remainder of the ow wasrecycled back into the bucket, aiding mixing of the sus-

pension in the bucket. The ow rate was measured bytiming the decrease in the level of the suspension in the

bucket. The ow rate used was approximately 13 cm3=s;this is suciently small for the ow to approximate that

of a pure plume at a short distance above the source. Thevolume of suspension injected into the tank during eachexperiment resulted in an increase of approximately 3%in the level of liquid in the tank. The eect of this changeon the ow in the plume and surroundings is sucientlysmall to be neglected in our modelling in Section 3. Theexperimental conditions for two nearly identical runs aresummarised in Table 1.

A number of trays were positioned radially away fromthe nozzle on the oor of the tank in order to collect thesedimenting particles. Each tray had a cross section of1:9 cm 1:9 cm. At the end of an experiment, the par-

ticles were extracted from each tray using a syringe andplaced in vials for analysis. The diameter of the samplingsyringe was 0:15 cm. The concentration of particles wasdetermined using a Coulter counter, which works on theelectrical sensing zone method (Allen, 1997).

2.2. Particles

The particles used in all experiments were sieved frac-tions of ballotini. These glass beads are nearly sphericaland suciently small to settle at low Reynolds number inwater. The Stokes settling velocity of an isolated particle


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S. S. S. Cardoso, M. Zarrebini/ Chemical Engineering Science 56 (2001) 47254736 4727

Table 1The experimental conditions

Experiment Flow rate Buoyancy ux Particle diameter Duration of Ambient density Particle concentrationQ0 (cm

3=s) B0 (cm4=s3) dp (m) experiment (s) e (g=cm3) at source C0 (g=l)

12a 13.24 203.4 74.38 506 1.0203 6.0016 12.86 215.8 73.03 476 1.0218 6.00

Fig. 2. The distribution of particle sizes in the source suspension.

was calculated for each diameterdi by

ui= d2i g

18(p e); (1)

where and e are the viscosity and density of water,

respectively, p is the density of the particle and g isthe acceleration of gravity. The local concentrations of

particles in the tank were typically smaller than 0.03%(v : v); these are suciently small for hindered settlingeects to be neglected (Batchelor, 1982; Huppert, Kerr,Lister, & Turner, 1991). The mean particle density is2:47 g=cm3. The particle-size distribution in each frac-tion was determined using the Coulter counter. The dis-tribution of particle sizes (by mass fraction) is shown inFig. 2. The distribution is bimodal, with particle diametersranging from 38 to 115m. Such distribution was usedto allow us to distinguish easily the behaviour of smaller

and larger particles in the system. The root mean squareparticle diameter for each experimental run is given inTable 1.

3. Model

Zarrebini and Cardoso (2000) have developed a rigor-ous model for the dynamics and deposition pattern fromaxisymmetric particle-laden plumes, composed of dense

particles of one size. Here we extend the model to accountfor the dispersion of sizes of the suspended particles.

Fig. 3. Schematic showing the trajectories of particles settling fromthe surface current.

3.1. Environment of innite lateral extent

Consider a turbulent, axisymmetric plume created by

the release of a buoyant suspension into a body of denserliquid of depth H. In general, the suspended particleswill be polydisperse with a range of settling velocities.We shall consider a discrete distribution of particle sizes,with a mass fraction fi of particles of Stokes settlingvelocityui fori = 1; : : : ; n. The concentration of particlesof typei in the suspension released at the source is thenC0i= C0fi, whereC0 is the concentration of particles atthe source; here the concentrations are expressed in mass

per unit volume of mixture.The plume rises to the surface of the liquid and spreads

out radially producing a turbulent surface current, as il-

lustrated in Fig. 3. The ow is suciently vigorous tomaintain a vertically uniform particle concentration inthe current. However, the particles of type i in suspen-sion, sediment across the bottom boundary of the surfacecurrent with their Stokes velocity ui (Martin & Nokes,1988; Sparks, Carey, & Sigurdson, 1991). The equationdescribing the transport of each type of particle in thecurrent is thus

QsdCi(rs) =2rsdrsCi(rs)ui; (2)

where rs denotes the radial position along the surfacecurrent,Qsis the volumetric ow rate of the plume at thefree surface andCi(rs) is the concentration of particles of


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type i at rs. Here, we have assumed that the volumetricow lost by sedimentation from the surface current isnegligible. Integration of Eq. (2) gives the concentrationof particles of type i along the surface current

Ci(rs) = Csiexp ui

Qs

(r2s b2

s ) ; (3)wherebsis the radius of the plume at the free surface andCsi is the concentration of particles of type i in the plumeatbs.

The path of a particular particle depends on its Stokesvelocity, as well as on the radial position at which it leavesthe surface current. Particles with large Stokes velocityor falling out of the current at small radii tend to bere-entrained into the plume, whilst particles with smallStokes velocities or falling out at large radial positionssettle on the ground. In general, the trajectory of a particleof typei is described by

dz

dr =

uiue

= uir

bw; (4)

whereueis the radial inward velocity of the environmen-tal uid (Zarrebini & Cardoso, 2000), b and w are theradius and velocity of the plume at height z, respectively,and = 0:125 is the entrainment constant (List, 1982;Turner, 1986).

Consider now two particles of types i and j. Particlei leaves the current at radius rsi , while particle j leavesthe current at radius rsj . Both particles are re-entrainedinto the plume at height zf. Then, using Eq. (4), we may

show that

uirsi drsi = ujrsj drsj : (5)

Conservation of particles of typei between the surfacecurrent and the edge of the plume requires that (see theappendix)

Ci(zf) =Ci(rsi )uirsi

bw cos2 i

drsidzf

; (6)

where i = arctan(ui=(w)). The radial inward velocity

at the edge of the plume, ue= w, is much greater thanthe Stokes velocities of the particles,ui (i= 1; : : : ; n), soi 0 (i= 1; : : : ; n). Hence,

Ci(rsi )uirsiCi(zf)

drsi=Cj(rsj )ujrsj

Cj(zf) drsj (7a)

= buedzf: (7b)

Combining Eqs. (5) and (7a) leads to

Cj(zf)

Ci(zf)=

Cj(rsj )

Ci(rsi ): (8)

The result in Eq. (8) simply states that the ratio of theconcentrations of dierent particles entering the plumeis equal to that leaving the surface current. But, fromEqs. (3) and (5)

Ci(rsi )

Cj(rsj )=

CsiCsj

exp

(b2 b2s )

(uj ui)

Qs

: (9)

Now, the velocity in the plume at the spreading level,ws = Qs=(b

2s ), is much greater than the Stokes velocities

of the particles, ui (i= 1; : : : ; n), so

(b2 b2s )(uj ui)

Qs1 (10)

and hence

Ci(rsi )

Cj(rsj )

CsiCsj

for particles i and j entering the plume at zf: (11)

Combining Eqs. (8) and (11), gives

Ci(zf)

Cj(zf)=

CsiCsj

: (12)

Eq. (12) shows that at each height zf, the ratio of theconcentrations of particles of types i and j entering the

plume is equal to the ratio of the concentrations of thesame particle types in the plume at the spreading level.

We may dene a critical radius rci for each particletype i, separating the region of re-entrainment from theregion of settling on the oor (Sparks et al., 1991). A

particle i settling from the current at rci has a trajectorywhich brings it back to the source of the plume, so that itis re-entrained (see Fig. 3). In steady state, the ow rateof particles i in the surface current at the critical radiusrci is equal to the ow rate of that type of particles at the

plume source

Q0C0i= QsCi(rci ); (13)

whereQ0 is the volumetric ow rate of suspension at thesource. And, combining Eqs. (11) and (13), we have

Csi

Csj=

C0i

C0j: (14)

Eqs. (12) and (14) show that at each height zf, the ratioof the concentrations of particles of types iandj enteringthe plume is equal to the ratio of the concentrations ofthe same particle types in the plume at the source level.Therefore, the ratio of concentrations of particle types iandj is constant in the plume.

Conservation of the mass of particles of type i, betweenthe surface current and the oor leads to

Ci(rf) =Ci(rsi )rsi

rf

drsidrf

: (15)


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Hence, using Eq. (5),

Ci(rsi )uirsidrsiCi(rf)

=Cj(rsj )ujrsjdrsj

Cj(rf) (16)

and

Ci(rf)

Cj(rf)= Ci(rsi )

Cj(rsj ) : (17)

This is a very powerful result in that it allows us to cal-culate the sedimentation prole on the oor for each par-ticle type j , given the prole for particle type i. Indeed,combining Eqs. (3), (14) and (17) yields

Ci(rf)

Cj(rf)=

C0iC0j

exp

(r2f b

2s )

uj uiQs

: (18)

The mass ow rate of particles of type j depositing onthe oor per unit radial distance is

Fj(rf) = 2rfCj(rf)uj : (19)

Substituting Eq. (18) into Eq. (19) gives

Fj(rf) = 2rfujCi(rf)C0jC0i

exp

(r2f b

2s )

ui uj

Qs

(20)

or

Fj(rf) =Fi(rf)C0j uj

C0i uiexp

(r2f b

2s )

ui ujQs

: (21)

The total mass ow rate of particles depositing on the

oor per unit radial distance is therefore

F(rf) =Fi(rf)

nj=1

C0j uj

C0i uiexp

(r2f b

2s )

ui uj

Qs

:

(22)

3.2. Dilute plumes in an innite environment

Zarrebini and Cardoso (2000) showed that if the con-centration of particles in the source suspension is small,such that

H

r 0

dedz1; (23)

then the motion of the plume is controlled by the buoy-ancy due to salinity decit; herer is the reference den-sity taken to be equal to the initial ambient density. Inthis case, the mass ux of particles of type i depositingon the oor is given by

Fi(rf) = 2rfuiCsiexp

uiQs

r2f b

2s +

Qsui

;

(24)

where

Csi=Q0C0i

Qsexp(1) (25)

and

Qs= 65

( 910

B0)1=3H5=3; (26)

whereB0is the buoyancy ux of the plume at the source.Using Eq. (22), the total mass ow of particles depositingon the oor per unit radial distance for a dilute plume isthus

F(rf) = 2rf

nj=1

Csj ujexp

(r2f b

2s )

uj

Qs 1

:

(27)

3.3. Environment of nite lateral extent

In an environment of nite lateral extent, the entrain-ment into the plume causes the environmental liquid sur-rounding the plume to move downward with an approxi-mately horizontally uniform velocity U(z). At the bottomof the radially spreading surface current, the liquid movesdownward with velocityUs and hence particles of type isettle from the current with vertical velocity Us+ ui. Amass balance of particles i in the surface current, takinginto account the radial decrease of volumetric ow rate,yields

Ci= Csi

1

UsQs

(r2 b2s )

ui=Us= Csi

R2 r2

R2 b2s

ui=Us:

(28)

The trajectory of a particle i leaving the surface currentis now described by

dz

dr =

ui+ U

ue=

(ui+ U)r

bw : (29)

Using Eq. (29), we may show that for two particles oftypes i andj, leaving the surface current at radial positionsrsi and rsj , respectively, and entering the plume at thesame levelzf or settling on the oor at radial positionrf,the following relation holds:

(ui+ Us)rsidrsi = (uj+ Us)rsjdrsj : (30)

Then, conservation of the mass of particlesi between thesurface current and the edge of the plume yields

Ci(zf)

Cj(zf)=

Ci(rsi )

Cj(rsj ) (31)


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Fig. 4. Theoretical predictions for the rate of deposition of sedimentin an environment of innite lateral extent.

and conservation of the mass of particles i between thesurface current and the oor leads to

Ci(rf)Cj(rf)

= Ci(rsi )Cj(rsj )

: (32)

Eqs. (31) and (32) show that the ratio of the concen-trations of dierent particles entering the plume, orsettling on the oor, is equal to that leaving the sur-face current. However, it is no longer possible to ndan analytical relation for the particle trajectories, andhence obtain analytical expressions for Ci=Cj(zf) and

Fig. 5. Particle ux at ground level as a function of particle diameter (innite environment), at (a) r= 3:4 cm, (b) r= 9:5 cm, (c) r= 17:6 cmand (d) r= 29:8 cm.

Ci=Cj(rf). The problem was therefore solved numeri-cally using Eqs. (31) and (32) in the code developed byZarrebini and Cardoso (2000), with new loops to accountfor a sequence of particle sizes.

In the next section, we present and compare experi-mental results with the theoretical predictions considered

above.

4. Results and discussion

In this section, we present experimental and theoreticalresults obtained using particles with the size distributionshown in Fig. 2. The new experimental results illustratethe eect of particle size on the deposition pattern. Theeects of buoyancy ux, particle concentration and tanksize on the sedimentation pattern from turbulent plumeshave been considered elsewhere (Zarrebini & Cardoso,2000; Cardoso & Zarrebini, 2001).

Fig. 4 shows the theoretical predictions for the rate ofaccumulation of sediment on the tank oor as a functionof radial distance, for an environment of innite lateralextent. Each solid curve shown corresponds to a spe-cic particle size. The larger particles deposit on the oorcloser to the plume source, whilst the settling of smaller

particles extends from the source to larger radial posi-tions. As a result, the rates of deposition of large particles


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Fig. 6. Root mean square particle diameter in the deposit as a functionof radial position (innite environment).

present well dened maxima at relatively small radial

positions, whilst the rates of settling of small particlesare more uniform throughout the tank oor. The dashedcurve shows the total rate of accumulation of sedimenton the oor.

Fig. 5(a)(d) shows the theoretical predictions andexperimental results for the rate of sedimentation of parti-cles of dierent sizes at given radial positions. The agree-ment between the experimental results and the theoretical

predictions is excellent. At small radial positions,r= 3:4and 9:5 cm, the ux of larger particles predominates overthe ux of smaller particles. However, further away fromthe plume source, the rate of sedimentation of smaller

particles becomes more signicant. At r= 29:8 cm, theux of smaller particles is larger than that of larger par-ticles. Experimentally, we nd that no large particles are

present at this radial position, possibly owing to the lowaccuracy of the measurements of very low particle con-centrations with the Coulter counter.

Fig. 6 shows the root mean square particle diameter inthe deposit as a function of radial position. The root meansquare diameter was chosen because the Stokes settlingvelocity of each particle is proportional to the square ofthe diameter of the particle. As expected, there is a steadydecrease in the root mean square particle diameter withincreasing radial position. The agreement between thetheoretical prediction (solid line) and the experimentalmeasurements is good, except at positions further awayfrom the plume, where the concentration of particles isvery low. At such low particle concentrations, the mea-surements are not accurate.

Fig. 7 presents the total rate of deposition in an inniteenvironment as a function of radial distance. The solidline represents the theoretical prediction taking into ac-count the polydispersivity of particles. The dashed linecorresponds to the theoretical prediction for a monodis-

perse particle fraction with root mean square particle di-ameter equal to that of the polydispersed particles. The

Fig. 7. Eect of the polydispersivity of particle sizes on the total rateof deposition in an environment of innite lateral extent.

Fig. 8. Comparison of the deposition patterns in environments ofnite and innite lateral extents.

new model including the polydispersion of particles isin better agreement with the experimental measurements.This gure illustrates that the polydispersivity of particleshas a signicant eect on the total deposition pattern, par-ticularly at larger radial positions. We note that at largeradial positions there is a noticeable dierence betweenthe experimental results for our two nearly identical runs;this dierence is attributed to the low accuracy of the con-centration measurements at low particle concentrations.

In Fig. 8, we show the eect of the lateral extent of thetank on the rate of settling of particles of dierent sizes.In a nite environment, the downward advection of theambient uid increases the particle ux and the maximumin the deposition rate lies closer to the source. This eectis more pronounced for the smaller particles sizes. Theeect of the nite size of the tank on the total rate ofdeposition is also shown. This eect, although alreadysignicant for the experimental conditions studied, will

be even more pronounced for larger buoyancy uxes atthe source of the plume (Zarrebini & Cardoso, 2000).

Fig. 9 presents the total rate of deposition in a nite en-vironment as a function of radial distance. The solid linerepresents the theoretical prediction taking into account


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Fig. 9. Eect of the polydispersivity of particle sizes on the total rateof deposition in an environment of nite lateral extent.

the polydispersivity of the particles. The dashed line cor-responds to the theoretical prediction for a monodisperse

particle fraction with equivalent root mean square par-ticle diameter. The predictions of the new model are ingood agreement with the experimental results. The simpleapproach considering an average particle diameter over-

predicts the particle ux just after the peak of depositionand underpredicts it at larger radial positions.

5. Application of the model to the emission ofchromium particles from a commercial plating plant

We shall now consider the application of the modeldeveloped in this paper to the discharge into the atmo-

sphere of polydispersed chromium particles from an ex-haust stack at a chromium plating plant.

Electroplating is the process of applying a metalliccoating to an object by passing an electric current throughan electrolyte. Hexavalent chromium baths are widelyused in industry to deposit chromium on metal objects.The baths consist of chromic acid, sulphuric acid andwater. The chromic acid is the source of the hexavalentchromium that deposits on the metal. The process evolveshydrogen and oxygen gases that bubble to the surface ofthe electrolyte. This results in the formation of a chromicmist, which is exhausted through an abatement process,including a cyclone separator and mesh pads, and thenreleased into the atmosphere.

Hexavalent chromium is a known carcinogen which,when inhaled, can cause several adverse health eects in-cluding runny noses, nose bleeds, ulcers, abdominal painand vomiting (US EPA, 1998a,b). More details on thehealth risks arising from exposure to chromium in the en-vironment are given on the US EPAs integrated risk in-formation system (IRIS). Reference concentration (RfC)for chronic inhalation exposure for chromium particulatesand chromic acid mist are included. They are estimatesof the daily inhalation exposure of the human populationthat is likely to be without an appreciable risk of harm-

Table 2Typical data for hexavalent chromium emissions from an exhauststack (Bonin et al., 1995; Bonin, 2000)

Exhaust stack

Height (m) 15Diameter (m) 0.5

Exit velocity (m=s) 2.5Exit temperature (K) 301Particulate emission rate (mg=s) 1:8 102

ful eects during a lifetime. The RfCs for chromic acidmists and hexavalent chromium particles are 8 106

and 1 104 mg=m3, respectively (US EPA, 1998a).Although the exhaust gases from a chromium plating

plant pass through an abatement process before being re-leased into the atmosphere, particles less than 1m indiameter, are emitted at a typical concentration of 3:5 102 mg=m3; this concentration is much larger than the

RfCs indicated above. It is therefore important to havea simple and easy to run model capable of predicting theconcentration levels of chromium particulates surround-ing the industrial source. The concentration proles ofthe dierent particle sizes at ground level will indicatewhich sizes are present in high proportions at given radial

positions. These are not necessarily the sizes present inhigh proportions at the source, for we have seen that size

plays a role in the ultimate deposition pattern. Our anal-ysis will indicate which particle sizes should be furtherremoved by an enhanced abatement process.

Bonin, Flower, Renzi, and Peng (1995) presented mea-

surements of the size and concentration of chromium par-ticles released from an electroplating facility in the UnitedStates. Particle samples were taken at three locations inthe exhaust stream: electroplating bath surface, exit of thecyclone separator and in the exhaust stack. Particle sizemeasurements were made with an Insitec particle counter,sizer, and velocimeter, capable of measuring particles di-ameters from 0:3 to 25m. In this paper, we use the dataof Bonin et al. (1995) to predict the deposition patternsof chromium particles released into the atmosphere fromthe exhaust stack (downstream of the abatement devices).The emission data used are presented in Table 2.

Our model is directly applicable to incompressible u-ids. In the example above, the vertical extent of motion inthe atmosphere will be such that it is not possible to treatair as an incompressible gas. However, the analysis maystill be applied to the atmosphere by replacing absolutetemperatures and densities for potential temperatures and

potential densities (Tritton, 1988). In a density stratiedenvironment, such as the atmosphere, the buoyancy uxin the plume decreases with height and the plume even-tually spreads out at the level at which the density in the

plume equals that in the atmosphere (the level of neutralbuoyancy). An equation for the height of rise of a plumein a linearly stratied environment was originally derived


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Fig. 10. The distribution of particle sizes of chromium in the exhauststack.

by Morton, Taylor, and Turner (1956) from a momen-tum, buoyancy and mass balance; uncertainties about themultiplying constant were later removed by direct exper-

iment (Briggs, 1969). We conne our approach to a stillatmosphere, with a positive, constant gradient of potentialtemperature in the vertical direction. The particle-ladengas will rise to a nite height Hgiven by (Turner, 1979)

H= 3:76B1=40 N

3=4: (33)

Here N is the buoyancy frequency, a measure of thestrength of the stratication, dened as

N=

g

1

d

dz

1=2=

g

Tr

1 +

d T=dz

1=2; (34)

where Tris the absolute temperature at the source leveland is the adiabatic temperature gradient.

We shall consider a standard atmosphere, for whichTr=288 K; =9:8 K=km and the rate of decrease of tem-

perature with height is 6:5 K=km (Morton et al., 1956).Thus,N= 0:02=s. We shall see that in the example un-der study, the buoyancy ux at the source is mainly dueto the dierence between the source temperature and theatmospheric temperature, as the particle concentration isso small that its contribution to the density of the plumeis negligible. Hence,

B0= g(T0 Tr)Q0; (35)

where T0 is the source temperature and = 1=Tr is thethermal expansion coecient of air. Once the height ofrise in the atmosphere is calculated from Eq. (33), weshall use the simplied model in Section 3.2 and thusneglect the eect of stratication on the plume behaviour.This simple analysis will suce to illustrate the behaviourof the release in a still atmosphere.

Fig. 10 shows the particle size distribution of hexava-lent chromium present in the exhaust stack investigated

by Bonin et al. (1995). The smallest particle size is 0:3min diameter (the lower measurable limit of the probe used

by those authors) and the largest is 0:8m. Particles withdiameters greater than 0:8m are only present upstreamof the abatement devices.

Fig. 11. Theoretical predictions for the total ux and concentrationof particles at ground level for a typical stack emission of chromiumparticles from a plating plant.

Fig. 12. Deposition rates for dierent particle sizes for a typicalchromium stack emission.

Eq. (33) predicts that the exhaust will rise to about44 m above the source. At this level the ow rate in the

plume is approximately 31 m3=s and the plume spreadsout radially forming a gravity current. Fig. 11 shows thetotal rate of deposition and the concentration of chromium

particles at the ground level. There is a pronounced max-imum in the rate of accumulation per unit radial distanceat approximately 800 m from the source. The maximumconcentration occurs near the source and is just below6 104 mg=m3. This value exceeds the RfC indicatedabove and hence further abatement measures might beconsidered. However, what is particle size range responsi-

ble for this relatively high concentration near the source?We answer this question below.

Fig. 12 shows the deposition rates for the dierent sizesof chromium particles. The larger particles deposit on theground closer to the plume source, up to radial positionsof 1500 m, whilst the settling of smaller particles extendsfrom the source to radii of 4000 m. As expected, therates of deposition of large particles exhibit well denedmaxima at positions less than 600 m from the source.

Fig. 13 shows the concentration prole of each particlesize at ground level. Near the source, particles of diameter


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Fig. 13. Ground concentration proles for dierent particle sizes fora typical chromium stack emission.

Fig. 14. Cumulative distribution of particle sizes at ground level, nearthe source, for a typical chromium stack emission.

0:41m are present in larger concentrations than anyother size, but at larger radial positions the concentrationof smaller particles becomes signicant.

Fig. 14 illustrates the cumulative distribution of particlesizes at ground level, near the source. We may concludethat to reduce the total concentration at this radial positionto the RfC of 1 104 mg=m3 indicated above, particlesof diameter greater than 0:40m should be removed byfurther abatement processes upstream from the exhaust.This removal will ensure that the total concentration ofchromium is below the RfC at all other radial positions.

As mentioned before, our experiments and model ig-nore the eects of both particleparticle interactions andwind on the particle sedimentation patterns. As a nalnote, it is instructive to discuss here the conditions underwhich our simplied analysis is valid. Particleparticleinteractions become important for particle concentra-tions larger than approximately 0:03% (v : v) (Batchelor,1982; Huppert et al., 1991). In the chromium emissionexample given above, the particle concentration at the

plume spreading level is approximately 1011% (v:v)and hence a given particle is so distant from its neigh-

bours that its motion is determined solely by the uid

motion. This assumption is not very restrictive and willbe valid in many problems involving industrial stackemissions (Zarrebini & Cardoso, 2000). On the otherhand, the eect of wind on the dispersal of particles can

be important (Woods et al., 1995); in the presence ofstrong wind, the plume will be carried away and will

have suered more entrainment by the time a givenheight is reached (Morton et al., 1956). We may expectthe transport of particles by wind to become signicantwhen the wind speed is of the same order or largerthan the radial inow driven by entrainment into theturbulent plume. In our chromium emission example,the entrainment velocity at the edge of the plume, nearthe spreading level, is approximately 0:3 m=s. There-fore, the results presented above are only valid for anatmosphere with very small wind speeds. This assump-tion becomes less restrictive as the buoyancy ux of the

plume increases due to a temperature increase and=or aow rate increase (see Eq. (35)).

6. Conclusions

We have presented analytical and numerical models forthe dynamics and deposition patterns of radially spread-ing gravity currents generated by axisymmetric turbulent

plumes. Our models take into account the settling of par-ticles of dierent sizes and=or densities. For simplicity,we describe here the particles with large Stokes settlingvelocities aslarge particlesand the particles with smallerStokes velocities as small particles.

A hierarchy of models was developed for increasinglycomplex situations. In the limit of dilute plumes, a sim-

ple analytical expression for the sedimentation prole onthe ground for each particle size was derived. In the caseof sedimentation in an environment of innite lateral ex-tent, we show that we can calculate the sedimentation

prole on the oor for any particle typej , given the pro-le for particle type i. This analytical nding was thenused in the numerical code to model the settling of poly-dispersed particles. Finally, we considered the sedimen-tation of polydispersed particles in a nite environment.This situation is more complex and we showed that itrequires full numerical solution.

Our theoretical predictions for the deposition patternson the oor were successfully compared with data fromlaboratory experiments. The large particles deposit onthe oor closer to the plume source, whilst the settlingof small particles extends from the source to larger ra-dial positions. At small radial positions, the ux of large

particles predominates over the ux of smaller particles.However, further away from the plume source, the rateof sedimentation of smaller particles becomes more sig-nicant. At suciently large radial positions, the ux ofsmaller particles is larger than that of larger particles.We nd that the root mean square particle diameter in


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the deposit decreases steadily with increasing radial po-sition.

We show that the eect of the polydispersivity of par-ticles on the total deposition pattern is signicant, par-ticularly at larger radial positions. The simple approachconsidering an equivalent root mean square particle di-

ameter overpredicts the particle ux just after the peak ofdeposition and underpredicts it at larger radial positions.In a nite environment, the downward advection of the

ambient uid increases the particle ux and the maxi-mum in the deposition rate lies closer to the source. Weshow that this eect is more pronounced for the smaller

particles sizes. The eect of the nite size of the tank onthe total rate of deposition is also more pronounced forlarger buoyancy uxes at the source of the plume.

We applied our new model to estimate the depositionpatterns of chromium particles exhausted from a stack at achromium plating plant in a still atmosphere. It was foundthat chromium particles with diameters less than 0:40m

are transported to distances up to 4000 m from the source,at ground level. The maximum ground concentration ofchromium particles for an emission with total particulateconcentration 3:5 102 mg=m3 is approximately 6 104 mg=m3 and occurs near the source. To reduce thisconcentration to a magnitude below the RfC indicated bythe US EPA, we showed that particles of diameter greaterthan 0:40m should be removed by further abatement

processes upstream from the exhaust.

Notation

b radius of the plume at height z; cmbs radius of the plume at the free surface, cm

B0 buoyancy ux in the plume at the source (atz= 0); cm4=s3

C0 concentration of particles at the source (atz= 0); g=cm3

C0i concentration of particles of type i at thesource (atz= 0); g=cm3

Ci(r) concentration of particles of type i at radialpositionr; g=cm3

Ci(zf) concentration of particles of type iat the edgeof the plume at heightzf; g=cm

3

Csi concentration of particles of type i in theplume atbs; g=cm

3

di diameter of particle typei; mfi mass fraction of particles of typei

Fi(rf) mass ow rate of particles of type i depositingalong the oor at rf per unit radial distance,g=cm s

g gravitational acceleration, cm=s2

H depth of environmental uid, cmN buoyancy frequency, s1

Q0 volumetric ow rate of suspension at thesource, cm3=s

Qs volumetric ow rate of the plume at the freesurface, cm3=s

rci critical radius for particle of typei, cmrf radial position along the oor, cmrs radial position along the surface current, cm

R equivalent radius of the environment, cm

T0 source temperature, KTr temperature in the environment at the sourcelevel, K

ui Stokes velocity of an isolated particle of typei, cm=s

ue radial inward velocity of the environmentaluid, cm=s

Us downward velocity in the environment at thebottom of the surface current, cm=s

U(z) downward velocity in the environment,cm=s

w velocity of the plume at height z, cm=sz; zf height above the source, cm

Greek letters

entrainment constant thermal expansion coecient of air, 1=K viscosity of water, g=cm s0 density of the suspension, g=cm

3

e density of water, g=cm3

p particle density, g=cm3

r reference density taken to be equal to theinitial ambient density, g=cm3

adiabatic temperature gradient, K=km

Appendix Conservation of particles between the surfacecurrent and the edge of the plume

In this appendix we derive Eq. (6). Consider theschematic in Fig. 15. A particle of type i leaves thesurface current with vertical velocity ui and is drawntowards the plume with radial velocity ue. At the edge

Fig. 15. Sketch of particle motion below the surface current.


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of the plume, the velocity of the particle is therefore u =u i+ u e.

Consider now an annulus of width drsi at the lowerboundary of the surface current. All particles settlingwithin this annulus will, at the edge of the plume, movewith velocity u through an annulus of width ds. Hence,

conservation of particles of type i between the surfacecurrent and the edge of the plume is given by

Ci(rsi )ui2rsidrsi = Ci(zf)u2b ds: (A.1)

From geometrical considerations (see Fig. 15)

u = uecos i

(A.2)

and

ds = dzf

cos i; (A.3)

where i = arctan (ui=ue). Substituting Eqs. (A.2) and(A.3) into Eq. (A.1), leads to

Ci(zf) =Ci(rsi ) uirsi

ueb cos2 i

drsidzf

: (A.4)

But from Eq. (4), at the edge of the plume,

ue=bw

b : (A.5)

Hence the concentration of particles of typei at the edgeof the plume at height zf is given by

Ci(zf) =Ci(rsi ) uirsi

bw cos2 i

drsidzf

: (A.6)

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