for a more complete understanding and subsequent ......a. dauriat and v. vesovic th huxley school of...

Modelling dispersion of heavy participate

matter

A. Dauriat and V. VesovicTH Huxley School of Environment, Earth Sciences and Engineering,

Imperial College, London SW7 2BP, UKE-mail: [email protected]

Abstract

This work examines the influence of the particle shape on heavy particledispersion in a neutrally stratified atmosphere. The particle trajectories areanalysed in terms of particle landing positions downwind from the source. Forthis purpose an already existing Lagrangian model [1] has been modified todeal with non-spherical particles. The particle trajectories are calculated bynumerically solving Newton's equations of motion, while the wind velocity ismodelled by means of a Markov chain scheme.

The simulation runs have been performed for lOOp, particles of differentnon-sphericity released from the elevated source. The results indicate that theshape of the particle is a very important parameter in determining thedeposition curves. In general with increasing non-sphericity, particles travelfurther from the source and the resulting ground-level particle distributionexhibits a larger spread.

The moments of particle deposition curves have been analysed and themedian of the distribution is related to the expected landing position of aparticle experiencing only average wind conditions.

1 Introduction

Atmospheric pollution arising from dispersion and deposition of paniculatematter is a growing concern because of its detrimental effect on human health,and the environment. In order to address this problem there is a genuine need

Transactions on Ecology and the Environment vol 29 © 1999 WIT Press, www.witpress.com, ISSN 1743-3541

844 Air Pollution

for a more complete understanding and subsequent quantification of thetransport of particles from the source downwind to the receptors. For heavyparticles, where the gravitational settling velocity cannot be neglected, it is notpossible to rely on models developed for gas dispersion, primarily due to twofactors. First the particle, because of its inertia, takes a finite amount of time torespond to the changes in the fluid velocity caused by turbulent fluctuations.Secondly, as the particle settles and drifts downwards under the influence ofgravity, it moves from one turbulent eddy to another, resulting in the 'crossing-trajectory effect'. Thus, it is not in general possible to equate particle velocitycorrelation functions with that of the surrounding fluid, nor is one fluid timescale sufficient to describe the relative particle motion.

The objective of this work is to develop a theoretically soundcomputational model of the heavy particle transport in the atmosphere buildingon the results of a number of workers who have addressed the problems ofparticle-fluid interactions [2-5]. Initially the model is to be used to conductparametric studies in order to ascertain the sensitivity of particle trajectories toparticle characteristics and meteorological conditions. Ultimately the model isenvisaged to be used to tackle the problems associated with the dispersion ofparti culate matter under conditions of high turbulence and complex terrain andto establish if such models are robust enough to be used for day-to-day analysis.

This work is the continuation of a previously reported study [1] whichdescribes the development of a two dimensional Lagrangian model for heavy,spherical particle dispersion from elevated sources under neutral stabilityconditions. The present work reports on the modifications of the model toincorporate the dispersion of non-spherical particles. It focuses on parametricsimulation studies performed in order to investigate the influence of theparticle shape on its dispersion.

2 Mathematical model

A two-dimensional mathematical model of dispersion of heavy particles inatmospheres under neutral stability conditions has been recently developed [1],The particle trajectory of a single, heavy particle in air is modelled by means ofNewton's equations of motion,

£-"•

where v, and %; are the components of the particle and wind velocityrespectively, and subscript i indicates either x or z-component. Both the particleand the fluid motion are viewed relative to the stationary frame of reference


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fixed in space in such a way that the negative z direction indicates the gravity

direction and z=0 indicates the ground level. The parameter i ( r - p^dl /TJ ) is

the particle aerodynamic response time, where p@ is the density of the particle,dy its equivalent volume diameter and TJ is the viscosity of air.

The factor/is the drag force correction factor. There have been a numberof attempts to represent the drag coefficient, CD, and consequently the dragforce correction factor /as a function of Reynold's number, Re. Although forspheres at least thirty equations have been proposed [6], for non-sphericalobjects the lack of experimental data and theoretical limitations have resultedin fewer attempts to describe the drag coefficient. In this work we haveemployed the recent general correlation proposed by Ganser [7] whichexpresses the drag coefficient of a non-spherical particle as a function of theReynolds number and variables that describe the particle shape. The drag forcecorrection factor is then given by an empirical expression,

. 3305 F/ = - - - - + 0.01794 Re*l+ , (3)

*

where parameters K\ and K̂ are Stokes' and Newton shape factors respectively[7]. In the Ganser correlation [7] they are related, by a set of empiricalexpressions which for brevity are omitted here, to the sphericity and projectedarea diameter for a given non-spherical particle. Provided the particle is ofgeometric shape, the sphericity, and the equivalent diameters can be calculatedwhich in turn would allow for the estimation of the K-shape factors and hencethe drag force correction factor, / by means of eqn (3), at any Reynold'snumber.

In order to compute the particle trajectory from its release to itsdeposition, eqns (1-2), one requires a knowledge of the wind components %%and %%. The wind velocity has been modelled by means of the Lagrangian typemodel as described previously [1]. The instantaneous wind velocity u has beendecomposed into two components namely the ensemble average wind velocity,IT ( %, ,0) and the turbulent fluctuation u' (w£ ,ŵ ). The average wind profile

%, (z) has been modelled by a well known log functionality [8]

- I n , ( 4 )K Z

where u is the friction velocity, ZQ is the surface roughness length and K is theVon Karman's constant equal to 0.4.

The turbulent fluctuations, u'^ and u^ have been simulated by means of a

simple Markov chain process as described previously [1,2,4,9], whereby theturbulent fluctuation, u' , at time t is given by


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I V2

077, (5)

CF, = 2.4w*, a, = 1.25w* (7)

where TJ is the Gaussian random number. The above formulation of the windvelocity has been relatively successfully in predicting experimentally observedwind profiles [9].

2.1 Numerical techniques

A set of four coupled first order stiff differential equations, eqns (1-2),describing the motion of a particle has been solved by a forward differencemethod based on variable time step and originally suggested by Gear [10]. Theadvantage of this method is that for the specified accuracy the time step isautomatically adjusted throughout the integration, thus leading to a largereduction in computation time.

The turbulent fluctuation components of the velocity field have beenevaluated from eqns (5-7). The Gaussian random numbers where generated bymeans of the minimal standard algorithms of Park and Miller [11] whichallowed for very large sequences of random numbers to be generated. This wasdeemed necessary since the previous use [1] of standard algorithms sometimesresulted in the random sequences required being larger than the pseudo-periodof the generator, thus invalidating the randomness of the numbers generated.During the simulations of large numbers of physically identical particles adifferent initial seed value was used in order to subject each particle to differentturbulent fluctuations.

2.2 Validation

The model presented by eqns (1-7) has been successfully tested for convergenceby means of simulating deposition of spherical particles as reported previously[1]. Furthermore the model predictions were compared with scant experimentaldata. For this purpose the field data of Hage [12], on releasing glassmicrospheres over a prairie, have been used. A good agreement between themodel predictions and the data was obtained [1]. Overall the model correctlypredicted the shape of the deposition curves and within 10% predicted theposition of the peak concentration, although the maximum concentration levelwas in a number of cases underestimated. Considering the simplicity of thepresent model and the uncertainty associated with the experimental data


Air Pollution 847

available the initial results reported [1] indicate that the proposed model iscapable of quantitatively describing the ground-level concentration ofpaniculate matter. Unfortunately, no validation was possible for non-sphericalparticles due to lack of any experimental field data.

3 Results

A number of simulations were run in order to ascertain the influence of particleshape on the resulting particle landing position distribution. The simulationshave been performed by releasing lOOp. equivalent-diameter particles ofnominal density of 2700 kg/m̂ from a height of 10m into average winds of5m/s. The average wind speed was specified at the height of 10m above groundand eqn (4) was used to compute the resulting wind profile. For this purposethe roughness length of zo=0.1m, corresponding to grassland, has been chosen.The results are presented as deposition curves in terms of distribution ofnormalized particle number frequency as a function of the landing distancedownwind from the source.

Before analysing the simulation runs it is instructive to examine theinfluence of the changes in the drag coefficient on particle deposition curves.For this purpose three different recommended correlations for the dragcoefficient of spherical particles were employed [6,7,13]. The first one istraditionally employed by chemical engineers [13] and in the transitionalregime it is based on the empirical correlation of Schiller and Naumann [14]developed in the 30's. The second correlation employed was developed recently

2.0E-02

'g 1.5E-02

l.OE-02

g 5.0E-03

O.OE4-00

0 50 100 150 200 250 300

downwind distance, m

Figure 1. Deposition curves for lOOji spherical particles for different dragcoefficient correlations (O [6]; D [7]; 0 [13, 14,]; ).


848 Air Pollution

by Haider, while the third correlation by Ganser [7] is described by eqn (3) ofthe present paper. Not surprisingly all three correlations give equivalent dragcoefficients in the Stokes' regime; with differences of up to 20% arising only atReynolds numbers corresponding to the transitional regime.

Figure 1 illustrates the effect of making use of a different drag coefficientcorrelation on the particle deposition curve for lOOp. spherical particles. Thelargest differences, of the order of 20-40%, are observed in the neighbourhoodof the maximum in the particle deposition curve. The resulting median andmean landing positions between the two recent drag coefficient correlations[6,7] vary by 10% and 15% respectively, while the difference between the twostandard deviations is of the order of 20%. This gives an illustration of thecurrent levels of uncertainty built in the models arising from the description ofparticle fluid interactions encountered in air dispersion.

In order to examine the influence of non-sphericity on particle landingpositions, the particles were divided into isometric and non-isometric shapes.

3.1 Isometric particles

A useful measure of the shape of isometric particles is the sphericity parameterand Ganser in his correlation [7] expressed both K-shape factors as empiricalfunctions of sphericity. The sphericity, (|>, is defined as the ratio of the surfacearea of a sphere of equivalent volume to the actual surface area of the particle.The values different from unity indicating departure from the spherical shape.Simulations were performed for non-spherical particles of three different

2.0E-02

1.5E-02

i&VH

I

l.OE-02

5.0E-03

O.OE+00

50 100 150 200 250


300

Figure 2. Deposition curves for isometric particles of different sphericity(— sphere; O

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sphericities, namely 0.91, 0.81 and 0.67, which approximately describe a cubeoctahedron, cube and tetrahedron respectively. All the simulations wereperformed for particles of equivalent volume diameter of lOO^i. Theconvergence runs, carried out for the different shapes, indicated that following3000 particle trajectories is sufficient for a stable particle deposition curve.

Figure 2 illustrates the influence of the shape of the isometric particle on theparticle deposition curve. As the sphericity parameter, , decreases and theshape is less spherical, the particles tend, on average, to travel further. This is adirect consequence of the increase of the drag coefficient as a function ofdecreasing sphericity, eqn (3). The peak concentration drops off dramaticallyand its position shifts towards larger downwind distances, as illustrated inFigure 2. The standard deviation of the deposition curves, measuring thespread of the distribution, increases with decreasing sphericity and for thelowest sphericity tested (tetrahedrons) the standard deviation of the depositioncurve is one and a half times larger than that of spherical particles.

3.2 Non-isometric particles

In order to describe the shape of the falling, non-isometric particles it is notsufficient to specify only the sphericity parameter [7]. In addition one needs tospecify the projected-area diameter, d*, which is a measure of the projected areaof the particle in its most stable position. Ganser [7] used both measures toobtain empirical expressions for the K-shape factors, K\ and K-I.

The non-isometric particles were modelled by means of disks of differentdimensions. The simulations were performed on four disks of differentsphericity and different projected area diameter. In order to compare withspherical and isometric particles the equivalent volume diameter of lOOp, waschosen for each disk and the sphericity of cj) = 0.91, 0.81, 0.67, 0.23 was usedrespectively. The choice of these two parameters defines the disk uniquely andone can then calculate the projected-area diameter for each disk to be used inestimating the K-shape factors of eqn. (3).

Figure 3 illustrates the influence of the shape of the non-isometric particleon the particle deposition curve. The peak concentration decreases withdecreasing sphericity and the peak position moves downwind from the source,but the changes are not as pronounced as for the isometric particles.Nevertheless as the sphericity decreases, the shape of the peak becomes lessGaussian and at the sphericity of =0.23 the deposition curve exhibits a verylong, flat tail-back. At this sphericity the disks are very thin with the ratio ofthe disk radius to its height approaching 20. Not surprisingly, such particlestake very much longer to settle and therefore travel much further downwindfrom the source, than the equivalent spherical particles.


850 Air Pollution

2.0E-02

'g L5E-02

Sf l.OE-02

j 5.0E-03

O.OE+00

0 50 100 150 200 250 300


Figure 3. Deposition curves for non-isometric particles of different sphericity(— sphere; O

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thus, supporting a reasonable conclusion that the 50% of particles will land atthe downwind distances smaller than a landing distance corresponding to thetransport by advection only. More importantly this allows for an easy estimateof the median of the Log-Normal Distribution, since the landing position of theparticle in the presence of advection only can be easily computed for anyshaped particle.

Table 1. Data on Log-normal distribution for selected particle shapes

Shape

spherecube oct.cube

tetrahedrondisk

Sphericity

*

1.000.910.810.670.23

Goodness-of-fit

t

0.01980.00640.01330.00540.0028

Medianx / m

66.272.379.890.6134.9

Landing position

jCadv./ m

66.171.577.887.8129.7

4 Conclusions

A model of heavy particle dispersion and deposition in a neutrally stratifiedatmosphere is presented. Influence of the particle shape on the deposition ofparticles was examined by means of the simulation runs performed forisometric and non-isometric lOOp, particles of different sphericity released fromthe elevated source.

The results indicate that the shape of the particle is a very importantparameter in determining the deposition curves. In general with increasingnon-sphericity, particles travel further from the source and the resultingground-level particle distribution exhibits a larger spread. The moments ofparticle deposition curves have been analysed and the median of thedistribution is related to the expected landing position of a particleexperiencing only average wind conditions.

References

1. Calviac, G. & Vesovic, V., Modelling dispersion of particulate matter inthe mining environment, Proc. of the 5* Fifth Int. Symp. onEnvironmental Issues and Waste Management in Energy and MineralProduction - SWEMP'98, eds. A.G. Pasamehmetoglu and A. Ozgenoglu,Balkema Publishers, Brookfield, pp!43-148, 1998.

2. Walkate, P.J., A random-walk model for dispersion of heavy particles inturbulent airflow, Boundary-Layer Meteorology. 39, pp. 175-190 1987,and references therein.


852 Air Pollution

3. Wang, L.P. & Stock, D.E., Numerical simulations of heavy particledispersion - scale ratio and flow considerations, Transactions ASME J. ofFluids Eng., 116, pp. 154-163, 1994.

4. Hashem, A. & Parkin, C.S., A simplified heavy particle random-walkmodel for the prediction of drift from agricultural sprays, AtmosphericEnvironment, 25A,, pp. 1609-1614, 1991.

5. Sawford, B.L. & Guest, P.M., Lagrangian statistical simulation of theturbulent motion of heavy particles, Boundary-Layer Meteorology, 54,pp. 147-166, 1991.

6. Haider, A. & Levenspiel, O., Drag coefficient and terminal velocity ofspherical and non-spherical particles, Powder Technology, 58, pp 63-70,1989.

7. Ganser, G.H., A rational approach to drag prediction of spherical andnon-spherical particles, Powder Technology, 77, pp 143-152, 1993.

8. Panofsky, H.A. & Button, J.A., Atmospheric turbulence. New York:John Wiley, 1984.

9. Hanna, S.R. Applications in air pollution modelling in atmosphericturbulence, Air pollution modelling, eds. FT. Nieuwstadt & H. van Dop,Dordrecht, D.Reidel, pp. 275-310, 1982.

10. Gear, C.W., Numerical initial value problems in ordinary differentialequations, Englewood Cliffs, N.J., Prentice-Hall, 1971.

11. Press, W.H., Teukolsky, S.A, Vetterling, W.T., & Flannery, B.P.,Numerical recipes; the art of scientific computing, Cambridge,Cambridge University Press, pp. 266-280, 1992.

12. Hage, K.D., On the dispersion of large particles form a 15-m source inthe atmosphere, J. Met. 18, pp. 534-539, 1961.

13. Coulson, J.M., Richardson, IF., Backhurst, J.R., &. Harker, J.H.,Chemical Engineering, 3"* edition, Vol., 2, Oxford, Pergamon Press, pp.86-122, 1980.

14. Schiller, L. and Naumann, A, Uber die grundlegenden Berechnungenbei der Schwerkraftaufbereitung, Z. Ver. deut. Ing. 77, pp318-330, 1933.

Acknowledgements

The authors would like to thank Guilliaume Bourtourault and Kate Dixon fortheir help in producing some of the figures in this paper.


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