350 Int. J. Environment and Pollution Vol. 5, Nos. 3 6, 1995

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1 350 Int. J. Environment and Pollution Vol. 5, Nos. 3 6, 1995 A puff-particle dispersion model P. de Haan and M. W. Rotach Swiss Federal Institute of Technology, GGIETH, Winterthurerstrasse 190, 8057 Zürich, Switzerland Abstract: A meso-scale puff diffusion model is presented which makes use of the concept of relative diffusion to disperse the puffs and moves their centres of mass by incorporating a Lagrangian stochastic dispersion model as it is normally used to describe particle dispersion. The so-called puff-particle model (PPM) retains the advantages of traditional puff models and at the same time is able to take into account the correct probability density function of the stochastic velocity components, with a computing time considerably less than that of particle models. The PPM is three-dimensional and it includes a Lagrangian approach for puff rise. The parametrizations cover stable, neutral and convective conditions. The model is shown to reproduce realistically the concentration patterns for a simple flow of homogeneous Gaussian turbulence. It is furthermore used to investigate the effect of an inhomogenity in surface roughness on dispersion characteristics. Keywords: dispersion, particle model, plume rise, puff model. Reference to this paper should be made as follows: de Haan, P., and Rotach, M.W. (1995) A puff-particle dispersion model, Int. J. Environment and Pollution, Vol. 5, Nos. 3 6, pp Introduction To take into proper account possible inhomogenities in the flow and turbulence fields when modelling pollutant dispersion, particle models (or random walk models) are generally accepted to be most appropriate. The major disadvantage of particle models, however, is their excessive need of computing time. At the other extreme, the relatively simple and fast plume models, based on one-particle statistics, yield a statistical plume, i.e. a concentration distribution that represents an ensemble average over a number of individual instantaneous plumes. This causes problems when modelling dispersion in a situation where the flow and/or turbulence field is inhomogeneous, with the scale of the inhomogenities not being symmetric with respect to the plume s axis or smaller then the plume itself. One way to circumvent this problem is the use of a puff approach with relative diffusion to disperse individual puffs. Because relative diffusion takes into consideration only the growth of a single puff, and not the average dispersion of many puffs, relatively diffused puffs can be used even over complex inhomogenities. However, in order to yield the proper spread of a plume described by many puffs (e.g., in the case of a continuously emitting source), that part of the turbulence spectrum that has no diffusive effect on the Copyright 1995 Inderscience Enterprises Ltd.

2 A puff-particle dispersion model 351 puff as a whole (i.e. the large eddies) must be included in the simulation, either by providing an updated wind-field every few seconds, or by artificially generating the effect of these eddies. Thus the idea behind the Puff-Particle Model (PPM) is to use the concept of relative diffusion to describe the spread of individual puffs as in other puff models (e.g. Mikkelsen et al., 1984), and additionally move their centres of mass using a stochastic particle model (Figure 1). With this, an ensemble average of puffs can be constructed (Figure ) without updating the meteorological fields too often. Furthermore, the effects of inhomogenity of the turbulence field can be taken into account, as it is the case in a traditional particle model. Figure 1 The concept of the PPM: The centres of mass of the individual puffs are moved along the trajectory generated by a particle model. Figure Sketch of the ensemble average of stochastically moved puffs, that are dispersed by relative diffusion, and correspond to a traditional plume. As mentioned earlier, the particle part of the PPM accounts for the effects of eddies being larger than the actual puff size and thus moving the puff as a whole. In consequence, the turbulence statistics in the particle model within the PPM should take into account only the low-frequency part of the spectrum, where the threshold frequency changes with time. For short times no large effect of this behaviour should be expected, but with increasing puff size important parts of turbulent fluctuations will have to be eliminated. However, at present, the particle model within the PPM still includes all eddy sizes in the calculation of the turbulence statistics. The puff approach is described in Section, and some details on the Lagrangian three-dimensional stochastic dispersion model are given in Section 3. The parametrizations of the turbulence statistics used in the PPM are briefly outlined in Section 4. To model the rise of polluted, warmer air within the concept of a Lagrangian puff model, a method to describe the rise of puffs without calculating an effective emission height, is presented in Section 5. As a first model validation the plume spread in homogeneous Gaussian turbulence, as obtained with the PPM, is compared to that of a Gaussian plume model (Section 6).

3 35 P. de Haan and M. W. Rotach A similar approach to combine the advantages of particle and puff models has recently been presented (Hurley, 1994). The vertical velocity components of his puffs are given by a one-dimensional Lagrangian particle model, whereas they are spread in the horizontal direction using a puff approach. In contrast to the PPM, Hurley s stochastic velocity components are used only to describe the vertical relative diffusion of the plume, and his model is therefore mainly suited to the description of dispersion in vertically inhomogeneous turbulence (e.g. a convective boundary layer). The puff model To describe the diffusion of a puff of particles (a cluster), the concept of relative diffusion (Richardson, 196) is adopted. Richardson emphasises that the rate of diffusion of two particles from each other is dependent on the distance of separation of the particles. This behaviour lies in the nature of turbulence, because only eddies smaller than the separation of two particles can contribute to an increase of this separation, i.e. are able to diffuse them. As the separation of two particles increases, larger eddies will diffuse them. Eddies that are too large to contribute to the diffusion will move both particles without diffusing them. Figure 3 shows the spreading of an ensemble averaged plume (modelled as a series of puffs using the sigma values according to one-particle or absolute diffusion characteristics), whereas Figure 4 shows the simulation of the corresponding plume by puffs dispersed with relative diffusion and additionally moved by large turbulent eddies. In the model, this additional movement is realised through the particle part of the PPM. Only a limited number of puffs are shown in Figure 4 for clarity and to highlight the size of puffs using the concept of relative dispersion (one individual plume ) compared with a statistical, ensemble-averaged plume (Figure 3). Figure 3 A series of puffs modelling a plume. Its spread is determined by one-particle or absolute diffusion. Source height 100 m.

4 A puff-particle dispersion model 353 Figure 4 Sizes of relatively diffused puffs from a source at a height of 100 m. Based on the work of Richardson, Smith and Hay (1961) derive an expression for the maximum rate of diffusion, which only depends on the intensity of turbulence, L = X 8 : dσ dx max βi, (1) where β is the ratio of the Lagrangian to the Eulerian time-scale. This expression should not be used for clusters, that are initially very small, because these clusters have a significant initial sub-normal rate of growth. Pasquill and Smith (1983) have further examined Equation 1 and obtained dσ dx max 0.3i. () Equation is used for the description of relative diffusion in other puff models as well, e.g., RIMPUFF (Mikkelsen et al., 1984). 3 The stochastic dispersion model To move the centres of mass of the puffs, a three-dimensional Lagrangian stochastic dispersion model is used that fulfils the well-mixed condition for stable/neutral to convective conditions. It is based on a model described in Rotach et al. (1995) (but is expanded to three dimensions). It allows a continuous transition between correlated Gaussian turbulence on one hand, and uncorrelated skewed turbulence on the other hand, of which the latter is characteristic for the convective boundary layer.

5 354 P. de Haan and M. W. Rotach The probability density function (PDF), P tot, describing the particle velocities, is composed of an uncorrelated part P conv = P u P v P ws and a correlated part P g = P u P v P w P uw, that are linked by a transition function, f : P tot = fp conv + ( 1 f)p g. (3) In P conv above, P ws is a skewed PDF in the vertical velocity, as described by Luhar and Britter (1989). From Equation 3 it can be seen that the total PDF becomes equal to P conv (i.e. all three velocity components are uncorrelated, w is skewed) if f =1. On the other hand, if f = 0, u and w are jointly Gaussian distributed ( P u P w P uw ) and the lateral velocity component 3 is assumed to be independent of the two others. From this PDF, the model Y for the velocity increments is constructed in the usual manner (see, e.g., Thomson, 1987). The transition function f is formulated in such a way that for large w * (the convective velocity scale) it becomes unity throughout most of the boundary layer and equal to zero over large parts of a neutral or stable boundary layer w * = 0 ( ) (Rotach et al., 1995). 4 Parametrization of turbulence statistics The PPM is designed for use under all stability conditions. For stable conditions the following parametrizations have been chosen: σ w u * = z z i 3, σ v u * = 1 z σ, u z i u * = 61 z, z i (4a c) that are adopted from Pasquill and Smith (1983), Gryning et al. (1987) and Stull (1988), respectively. For neutral to convective conditions, parametrizations are used which combine the limits for neutral stability and convective conditions, σ u w * σ v u * σ u u * 3 =1.5 z exp z z z i z i z i = 0.35 z 3 i kl + z z i = 0.35 z 3 i kl z. z i u * w * (5a) (5b) (5c) as proposed by Pasquill and Smith (1983) and Rotach et al. (1995). Note that the neutral limits of the Equations 5a c do not correspond those of Equations 4a c.

6 A puff-particle dispersion model Puff rise In order to model the rise of an exhaust gas warmer than the surrounding air ( plume rise ), Briggs (1975) formulas have often been used. These expressions were derived for plume models and it is difficult to make use of them within a Lagrangian framework. Therefore, the source height is often replaced by an effective stack height for each puff. In the following, a Lagrangian approach to calculate puff rise is presented and briefly discussed. The principles on which the calculation of the puff rise is based, are briefly outlined in the following. Mainly, two forces have an effect on a parcel of air: the buoyancy and the air resistance. Additionally, a parcel of air will exchange heat with the surrounding air. The heat flux, I, through the surface area, A, of the parcel is I = λ T A x, (6) where λ denotes the thermal conductivity and [ is the distance over which the maximum temperature difference 7 is measured. The buoyancy of a parcel of air with mass m is F A = g m m * ρ T p R, (7) where J = 9.8ms, 5 = 8.134J K 1 mol 1 and p is the air pressure. In Equation 7, the density ρ and the mass per mole P are set equal for the parcel and the surrounding air. Furthermore, while rising with a velocity w, a parcel of air experiences a frictional force, F W, F W = c W ρw B, (8) where B is the area normal to the wind velocity and c W is the drag coefficient. During a time step W the parcel of air experiences a vertical acceleration a = ( ) F + F A W m, (9) but owing to the heat flux described by Equation 6 the temperature surplus according to the surrounding air is continuously reduced. At each instant after release, a puff s tendency to rise can be examined by evaluating Equation 9. Unfortunately, a puff is not a parcel of air, but rather a probability density function of a pollutant concentration. Thus c W in Equation 8 can be quite different from the corresponding value for a sphere, say. Furthermore, compared with a parcel with a closed surface, a puff has a far better exchange with ambient air, which causes an increase in the heat conductivity λ. In fact, the determination of the parameters c W and λ is the trickiest part of this new approach to describe the rising of warm air (puff rise). So far, for the use within the PPM, there stability-dependent sets of values have been determined by a parameter fit enforcing exactly the same results as by using Brigg formulas for plumes. Clearly, it would be desirable to make direct use of experimental data for a definite parametrization of c W and λ. Figure 5 shows the rise of a plume simulated by individual puffs, that reach their final height after about 100 m. On the ordinate the effective emission height after Briggs

7 356 P. de Haan and M. W. Rotach formulas for neutral stability is indicated. The values used for the parameters F and λ are : those for neutral stability. Figure 5 Rise of puffs with use of the puff-rise procedure. The puffs are initially 60 C warmer than the environment. The mean wind speed is 4 ms 1. The arrow B indicates the effective emission height after Briggs (1975). 6 Results The performance of the PPM is investigated by analysing a case of homogeneous, Gaussian turbulence. Under these conditions, a Gaussian (statistical) plume can be viewed as a reference. Thus the ability of the PPM is investigated to reproduce realistically a statistical plume by ensemble-averaging over a number of puffs that experience a stochastic velocity to move their centre of mass. Figure 6 shows the crosssection of such a statistical plume. For comparison, Figure 7 shows the corresponding plume, as obtained by relatively diffused and stochastically moved puffs. The concentration isolines to the values and show a satisfactory resemblance. The differences at the edges of the plume (i.e., the different positioning of the and contour-lines) are due to the fact that the sigma values of the relatively diffused puffs are smaller than those used to describe a statistical plume. This leads to differences at very low concentrations at the outer regions of a plume, which are of minor physical relevance, though.

8 A puff-particle dispersion model 357 Figure 6 Cross-section of a Gaussian plume at x = 77.4 m. Source height, 100 m. The sigma values describing the plume are those of stability class D. Figure 7 Cross-section at x = 77.4 m of a plume simulated by the PPM under neutral stability conditions.

9 358 P. de Haan and M. W. Rotach As an example for the ability of the PPM to simulate the dispersion of pollutants over a surface inhomogenity, a roughness strip with ] = 1 0 m within smooth terrain (] = m) is examined. Figure 8 shows the simulation with the PPM, whereas in Figure 9 the result for a Gaussian plume model is depicted, where the same source has been used in both cases. Figure 8 Concentration isolines (vertical cross-section at z = 0 m) of a plume released at (0, 0, 0 m), modelled by the PPM. The strip between x = 400 and x = 600 m has a roughness length 50 times larger than the surrounding terrain. The wind and turbulence field for the PPM was provided by a simulation with a k-ε model. For the plume model, the equilibrium upwind conditions of this simulation were used as input. It can be seen that the spread within the plume as modelled with the PPM increases as soon as the strip with the enhanced roughness is reached (Figure 8). For the case of a surface release closer to the roughness strip, the difference between the plume model and the PPM can be expected to be even larger than for the case depicted in Figures 8 and 9.

10 A puff-particle dispersion model 359 Figure 9 The same situation as in Figure 8, but modelled with a Gaussian plume model. 7 Acknowledgements Many thanks go to D. Bünzli, GGIETH, for doing the simulation of the k-ε wind-field. References Briggs, G.A. (1975) Plume rise predictions, in Lectures on Air Pollution and Environmental Impact Analyses, AMS, Boston, pp Gryning, S.-E., Holtslag, A.A.M., Irwin, J.S. and Sivertsen, B. (1987) Applied dispersion modelling based on meteorological scaling parameters, Atmospheric Env., Vol. 1, pp Hurley, P. (1994) Partpuff A Lagrangian particle-puff approach for plume dispersion modelling applications, Appl. Meteorol., Vol. 33, pp Luhar, A.K. and Britter, R.E. (1989) A random walk model for dispersion in inhomogeneous turbulence in a convective boundary layer, Atmospheric Env., Vol. 3, pp Mikkelsen, T., Larsen, S.-E. and Thykier-Nielsen, S. (1984) Description of the Risø puff diffusion model, Nuclear Technology, Vol. 67, pp Pasquill, F. and Smith, F.B. (1983) Atmospheric Diffusion, 3rd edn, John Wiley & Sons, New York. Richardson, L.F. (196) Atmospheric diffusion shown on a distance-neighbour graph, Proc. Royal Soc., Series A, Vol. 110, pp Rotach, M.W., Gryning, S.-E. and Tassone, C. (1995) A two-dimensional Lagrangian stochastic dispersion model for daytime conditions, submitted to Quart. J. Roy. Meteorol. Soc. (Copies are available from the authors of this paper.) Smith, F.B. and Hay, J.S. (1961) The expansion of clusters of particles in the atmosphere, Quart. J. Roy. Meteorol. Soc., Vol. 87, pp Stull, R.B. (1988) An Introduction to Boundary Layer Meteorology, Kluwer Academic Publishers, Dordrecht. Thomson, D.J. (1987) Criteria for the selection of stochastic models of particle trajectories in turbulent flows, J. Fluid Mech., Vol. 180, pp

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