Modelling Dispersion of Heavy Participate Matter

Modelling dispersion of heavy participate matter

A. Dauriat and V. Vesovic

TH Huxley School of Environment, Earth Sciences and Engineering, Imperial College, London SW7 2BP, UK

Abstract

This work examines the influence of the particle shape on heavy particle dispersion in a neutrally stratified atmosphere. The particle trajectories are analysed in terms of particle landing positions downwind from the source. For this purpose an already existing Lagrangian model [1] has been modified to deal with non-spherical particles. The particle trajectories are calculated by numerically solving Newton's equations of motion, while the wind velocity is modelled by means of a Markov chain scheme.

The simulation runs have been performed for lOOp, particles of different non-sphericity released from the elevated source. The results indicate that the shape of the particle is a very important parameter in determining the deposition curves. In general with increasing non-sphericity, particles travel further from the source and the resulting ground-level particle distribution exhibits a larger spread. The moments of particle deposition curves have been analysed and the median of the distribution is related to the expected landing position of a particle experiencing only average wind conditions.

1 Introduction

Atmospheric pollution arising from dispersion and deposition of paniculate matter 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

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for a more complete understanding and subsequent quantification of the transport of particles from the source downwind to the receptors. For heavy particles, where the gravitational settling velocity cannot be neglected, it is not possible to rely on models developed for gas dispersion, primarily due to two factors. First the particle, because of its inertia, takes a finite amount of time to respond to the changes in the fluid velocity caused by turbulent fluctuations.

Secondly, as the particle settles and drifts downwards under the influence of gravity, it moves from one turbulent eddy to another, resulting in the 'crossing- trajectory effect'. Thus, it is not in general possible to equate particle velocity correlation functions with that of the surrounding fluid, nor is one fluid time scale sufficient to describe the relative particle motion. The objective of this work is to develop a theoretically sound computational model of the heavy particle transport in the atmosphere building on the results of a number of workers who have addressed the problems of particle-fluid interactions [2-5]. Initially the model is to be used to conduct parametric studies in order to ascertain the sensitivity of particle trajectories to particle characteristics and meteorological conditions. Ultimately the model is envisaged to be used to tackle the problems associated with the dispersion of parti culate matter under conditions of high turbulence and complex terrain and to 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] which describes the development of a two dimensional Lagrangian model for heavy, spherical particle dispersion from elevated sources under neutral stability conditions. The present work reports on the modifications of the model to incorporate the dispersion of non-spherical particles. It focuses on parametric simulation studies performed in order to investigate the influence of the particle shape on its dispersion.

2 Mathematical model

A two-dimensional mathematical model of dispersion of heavy particles in atmospheres under neutral stability conditions has been recently developed [1],

The particle trajectory of a single, heavy particle in air is modelled by means of Newton's equations of motion,

£-"•

where v, and %; are the components of the particle and wind velocity respectively, and subscript i indicates either x or z-component. Both the particle and 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 number of attempts to represent the drag coefficient, CD, and consequently the drag force correction factor /as a function of Reynold's number, Re. Although for spheres at least thirty equations have been proposed [6], for non-spherical objects the lack of experimental data and theoretical limitations have resulted in fewer attempts to describe the drag coefficient. In this work we have employed the recent general correlation proposed by Ganser [7] which expresses the drag coefficient of a non-spherical particle as a function of the

Reynolds number and variables that describe the particle shape. The drag force correction factor is then given by an empirical expression,

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

where parameters K\ and K^ are Stokes' and Newton shape factors respectively [7]. In the Ganser correlation [7] they are related, by a set of empirical expressions which for brevity are omitted here, to the sphericity and projected area diameter for a given non-spherical particle. Provided the particle is of geometric shape, the sphericity, and the equivalent diameters can be calculated which in turn would allow for the estimation of the K-shape factors and hence the drag force correction factor, / by means of eqn (3), at any Reynold's number.

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

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

-In, (4) K Z

where u is the friction velocity, ZQ is the surface roughness length and K is the Von 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 the turbulent 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 wind

velocity has been relatively successfully in predicting experimentally observed wind 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 difference

method based on variable time step and originally suggested by Gear [10]. The advantage of this method is that for the specified accuracy the time step is automatically adjusted throughout the integration, thus leading to a large reduction in computation time.

The turbulent fluctuation components of the velocity field have been evaluated from eqns (5-7). The Gaussian random numbers where generated by means of the minimal standard algorithms of Park and Miller [11] which allowed for very large sequences of random numbers to be generated. This was

deemed necessary since the previous use [1] of standard algorithms sometimes resulted in the random sequences required being larger than the pseudo-period of the generator, thus invalidating the randomness of the numbers generated.

During the simulations of large numbers of physically identical particles a different initial seed value was used in order to subject each particle to different turbulent fluctuations.

2.2 Validation

The model presented by eqns (1-7) has been successfully tested for convergence by means of simulating deposition of spherical particles as reported previously

[1]. Furthermore the model predictions were compared with scant experimental data. For this purpose the field data of Hage [12], on releasing glass microspheres over a prairie, have been used. A good agreement between the model predictions and the data was obtained [1]. Overall the model correctly

predicted the shape of the deposition curves and within 10% predicted the position of the peak concentration, although the maximum concentration level was in a number of cases underestimated. Considering the simplicity of the present model and the uncertainty associated with the experimental data

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available the initial results reported [1] indicate that the proposed model is

capable of quantitatively describing the ground-level concentration of paniculate matter. Unfortunately, no validation was possible for non-spherical particles due to lack of any experimental field data.

3 Results

A number of simulations were run in order to ascertain the influence of particle shape on the resulting particle landing position distribution. The simulations have been performed by releasing lOOp. equivalent-diameter particles of

nominal density of 2700 kg/m^ from a height of 10m into average winds of 5m/s. The average wind speed was specified at the height of 10m above ground and eqn (4) was used to compute the resulting wind profile. For this purpose the roughness length of zo=0.1m, corresponding to grassland, has been chosen.

The results are presented as deposition curves in terms of distribution of normalized particle number frequency as a function of the landing distance downwind from the source. Before analysing the simulation runs it is instructive to examine the

influence of the changes in the drag coefficient on particle deposition curves. For this purpose three different recommended correlations for the drag coefficient of spherical particles were employed [6,7,13]. The first one is

traditionally employed by chemical engineers [13] and in the transitional regime 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 drag coefficient correlations (O [6]; D [7]; 0 [13, 14,]; ).

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

Reynolds numbers corresponding to the transitional regime. Figure 1 illustrates the effect of making use of a different drag coefficient correlation on the particle deposition curve for lOOp. spherical particles. The largest differences, of the order of 20-40%, are observed in the neighbourhood of the maximum in the particle deposition curve. The resulting median and mean landing positions between the two recent drag coefficient correlations [6,7] vary by 10% and 15% respectively, while the difference between the two standard deviations is of the order of 20%. This gives an illustration of the current levels of uncertainty built in the models arising from the description of particle fluid interactions encountered in air dispersion. In order to examine the influence of non-sphericity on particle landing positions, 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 parameter and Ganser in his correlation [7] expressed both K-shape factors as empirical functions of sphericity. The sphericity, (|>, is defined as the ratio of the surface area 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 l.OE-02 & VH

5.0E-03 I

O.OE+00 50 100 150 200 250 300

downwind distance, m

Figure 2. Deposition curves for isometric particles of different sphericity (— sphere; O<&=0.91; A<|>=0.81 0 (jp=0.67;).

Air Pollution 849 sphericities, namely 0.91, 0.81 and 0.67, which approximately describe a cube octahedron, cube and tetrahedron respectively. All the simulations were performed for particles of equivalent volume diameter of lOO^i. The convergence runs, carried out for the different shapes, indicated that following 3000 particle trajectories is sufficient for a stable particle deposition curve.

Figure 2 illustrates the influence of the shape of the isometric particle on the particle deposition curve. As the sphericity parameter, <(>, decreases and the shape is less spherical, the particles tend, on average, to travel further. This is a direct consequence of the increase of the drag coefficient as a function of decreasing sphericity, eqn (3). The peak concentration drops off dramatically and its position shifts towards larger downwind distances, as illustrated in Figure 2. The standard deviation of the deposition curves, measuring the spread of the distribution, increases with decreasing sphericity and for the lowest sphericity tested (tetrahedrons) the standard deviation of the deposition curve 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 not sufficient to specify only the sphericity parameter [7]. In addition one needs to specify the projected-area diameter, d*, which is a measure of the projected area of the particle in its most stable position. Ganser [7] used both measures to obtain empirical expressions for the K-shape factors, K\ and K-I.

The non-isometric particles were modelled by means of disks of different dimensions. The simulations were performed on four disks of different sphericity and different projected area diameter. In order to compare with spherical and isometric particles the equivalent volume diameter of lOOp, was chosen for each disk and the sphericity of cj) = 0.91, 0.81, 0.67, 0.23 was used respectively. The choice of these two parameters defines the disk uniquely and one can then calculate the projected-area diameter for each disk to be used in estimating the K-shape factors of eqn. (3). Figure 3 illustrates the influence of the shape of the non-isometric particle on the particle deposition curve. The peak concentration decreases with decreasing 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 less Gaussian and at the sphericity of <|>=0.23 the deposition curve exhibits a very long, flat tail-back. At this sphericity the disks are very thin with the ratio of the disk radius to its height approaching 20. Not surprisingly, such particles take very much longer to settle and therefore travel much further downwind from the source, than the equivalent spherical particles.

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2.0E-02

'g L5E-02

Sf l.OE-02

j 5.0E-03

O.OE+00

0 50 100 150 200 250 300 downwind distance, m

Figure 3. Deposition curves for non-isometric particles of different sphericity (— sphere; O<|F=0.91; 0

3.3 Deposition curves

The deposition data illustrated in figures 2 and 3 are useful estimates of particle ground-level concentration downwind from the source, but their generation does require sometimes lengthy computational simulations. This in fact sets the lower limit on the size of the particles one can consider by particle trajectory methods. This is one of the reasons why it would be desirable to represent the deposition data by a known distribution function, specially if the moments of the function could be related to some particle and meteorological characteristics. For this purpose, some preliminary work has been carried out, whereby the deposition histograms obtained where fitted to Log-Normal Distribution. The Log-Normal distribution is a relatively simple distribution that requires the specification of only two moments, namely the median and the standard deviation, for completeness. Furthermore statistical data on a number of processes in nature seems to follow it. The results of the fitting procedure reported as goodness-of-fit in Table 1 and visual inspection indicate that a reasonable degree of fit has been obtained. Table 1 also summarizes the median of the fitted distribution for a number of selected shapes. It is instructive to compare the median presented in Table 1 for each simulation with the respective landing position, %,%&., of the particles which are not subjected to any turbulence, but are advected by the average wind profile alone. The agreement is good and the largest difference of 4% is observed at the lowest sphericity,

Air Pollution 851 thus, supporting a reasonable conclusion that the 50% of particles will land at the downwind distances smaller than a landing distance corresponding to the transport by advection only. More importantly this allows for an easy estimate of the median of the Log-Normal Distribution, since the landing position of the particle in the presence of advection only can be easily computed for any shaped particle.

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

Shape Sphericity Goodness-of-fit Median Landing position

t x / m jCadv./ m *

sphere 1.00 0.0198 66.2 66.1 cube oct. 0.91 0.0064 72.3 71.5

cube 0.81 0.0133 79.8 77.8 tetrahedron 0.67 0.0054 90.6 87.8

disk 0.23 0.0028 134.9 129.7

4 Conclusions

A model of heavy particle dispersion and deposition in a neutrally stratified atmosphere is presented. Influence of the particle shape on the deposition of particles was examined by means of the simulation runs performed for isometric and non-isometric lOOp, particles of different sphericity released from the elevated source. The results indicate that the shape of the particle is a very important parameter in determining the deposition curves. In general with increasing non-sphericity, particles travel further from the source and the resulting ground-level particle distribution exhibits a larger spread. The moments of particle deposition curves have been analysed and the median of the distribution is related to the expected landing position of a particle experiencing only average wind conditions.

References

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Acknowledgements

The authors would like to thank Guilliaume Bourtourault and Kate Dixon for

their help in producing some of the figures in this paper.