Analytical air pollution modeling T. Tirabassi ABSTRACT

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1 Analytical air pollution modeling T. Tirabassi ABSTRACT In this paper we present some advanced practical models that use solutions of the advection-diffusion equation that accept wind and eddy diffusivity profiles that are power functions of the height. The performance of these models has been assessed with success in cases of both ground level sources as well as elevated sources. INTRODUCTION Atmospheric-dispersion modelling usually takes accounts of atmospheric turbulence by means the method based on Pasquill-Gifford classes. These are utilised where the meteorological state of the atmospheric boundary layer is classified in a simple way based on surface measurements, and where the dispersion from a source is estimated by assuming simple formulae for concentration distribution, in which the dispersion parameters depend simply on downwind distance and the meteorological state of boundary layer. In the summary by Weil [1] of the American Meteorological Society and United State Environmental Protection Agency Workshop is suggested that similarity approach replaces the classical Pasquill-Gifford stability classification. Experimental work and modelling efforts have attempted to parameterized the surface fluxes of momentum, heat and moisture in terms of routinely measured meteorological parameters (e.g. Holtslag and van Ulden [2]; van Ulden and Holtslag 3]; Trombetti et al. [4]; Beljaars and Holtslag [5]). Various organisation word-wide are introducing advanced modelling techniques that make use of the above recent researches on the meteorological state of the boundary layer. These advanced modelling techniques (van Ulden [6]; Berkowicz et al. [7]; Tirabassi [8]; Hanna and Paine [9]; Carruthers [10]) contain algorithms for calculating the main factors that determine air pollution diffusion in terms of the fundamental

2 524 Computer Simulation parameters the Monin-Obukhov length scale. Within this framework, we have developed Several model codes that employ an analytical solution which is not Gaussian but it allows for variations in the wind and exchange coefficients with height. ANALYTICAL SOLUTION OF THE ADVECTION-DIFFFUSION EQUATION Roberts [11] presented a bidimensional solution, for ground level sources only, in cases where both the wind speed (u) and vertical diffusion coefficients (K, ) follow power laws as a function of height (z). That is: u = u, (z/z, ) = K, = K, (z/z, / where z, is the height where u, and K, are evaluated. Rounds [12] obtained a bidimensional solution valid for elevated sources, but only for linear profiles of K.. Smith [13] resolved the bidimensional equation of transport and diffusion with u and K. power functions of height with the exponents of these functions following the conjugate law of Schmidt (that is: 'wind exponent' = l-'k. exponent'). Smith [14] also presented a solution in the case of constant u, but K. following: ^=^*(#-z)* where K^isa constant and a and b can be: a > 0 and b = 0 a = 0 and b > 0 for 0 < z < H a = 1 and b > 0 for 0 < z < H a = 1 and b = 0 for 0 < z < H/2 ; a = 0 and b = 1 for H/2 < z < H where H is the height of the atmospheric boundary layer. Scriven and Fisher [15] proposed a solution with constant u and K. as: K, = z for 0 < z < z, K, = K, (z, ) for z, < z < H where z, is a predetermined height (generally, the height of the surface layer). This solution allows (as boundary conditions) a net flow of material towards the ground: where V^ is the deposition velocity. The solution of Scriven and Fisher has been widely used in the United Kingdom for long range transport of pollutant.

3 Computer Simulation 525 Yeh and Huang [16] and Berlyand [17] published bidimensional solutions for elevated sources with u and K. following power profiles, but for a unbound atmosphere. That is: K, ^ =0 atz = QO dz Demuth [18] put forward a solution with the same conditions, but for a vertically limited boundary layer. That is: *\^1 K. =0 atz = H By applying the Monin-Obukhov similarity theory to diffusion, van Ulden [6] derived a solution for vertical diffusion from continuous sources near the ground only with the assumption that u and K. follow power profiles. His results are similar to Roberts', but he provided a model for nonground level sources, but applicable to sources within the surface layer. Nieuwstadt [19] presented a solution which was a particular case of Smith's [14] solution noted above. Subsequently, Nieuwstadt and Haan [20] extended that solution to the case of a growing boundary layer height. Catalano [21], in turn, extended the latter solution to the case of non-zero mean vertical wind profiles. ANALYTICAL MODELS We have developed some models that utilise the two dimensional analytical solutions of the advection diffusion equation proposed by Yeh and Huang [16] and Berlyand [17] for a unbounded boundary layer and by Demuth [18] for a bounded one, while the cross-wind dispersion is simulated by a Gaussian term. That is: c\, C(x,y.z) = /= exp v27rcr. y~ (1) where Cy is the cross-wind integrated concentrations, y is the cross-wind axes and c^ is the lateral diffusion parameter. The solutions of Yeh and Huang [16] and Berlyand [17] can be written: C - while the Demuth' solutions [19] is:

4 526 Computer Simulation C =./%-'(&(,)) exp - where x is the along-wind direction, Q the source emission, h the source effective height, H the mixing height, A = a + /?-2, v=(l-/?)/2, r = (a+l)/a, r = p-a, R = h/h, p = (l-/?)/2, q = A/2, J, and I, represent the Bessel function and modified Bessel function of first kind and order y and v, p^ the roots of J^. Generally the models parameterize the wind profiles approximating actual profiles with power low which preserve the speed at the effective source height and the integral of the profile between ground level and effective height. That is, in mathematical terms: u/h) = u, (h) where u^is the power law fitting and u^ is the observed profile. From the above condition a single formula for the exponent of power law wind profile can be obtained: Similar conditions for the diffusion coefficient profile give: (4) If the effective height source h is less than the surface layer height then the latter is used instead of h. We have set up four different analytical model based on the above solutions: KAPPAG, KAPPAG-LT, CISP and MAOC.

5 Computer Simulation 527 The KAPPAG model The model can handle multiple sources and multiple receptors, simulating time-varying conditions in which each time interval (e.g., 1 hour) is treated as a stationary case. The model output is a statistical summary of the concentrations computed at each receptor, during each time step, and due to each source. Partial and total concentrations are computed for hourly and multi-hour averages. Highest and second- highest values are also evaluated. The performances of KAPPAG model has been assessed with success by comparing ground level concentration estimates with SO, relatively to ground level releases (Tagliazucca et al. [23] and with SF^ elevated releases (Tirabassi et al. [24]). data relative to The KAPPAG-LT model KAPPAG-LT is the long term version of KAPPAG and it estimates annual ground level concentrations. Its performance has been assessed with success against SO, data from heavily industrialised area (Tirabassi et al. [25]). The CISP model CISP (Tirabassi and Rizza, [26]) is a screen model that provides a method for estimating maximum ground level concentrations from a single point source as a function of stability and wind speed. In fact, it is designed for the lowcost, detailed screening of point sources to determine maximum one-hour concentrations and to decide whether use of one of the more sophisticated models is required. CISP model is regarded as a useful tool for a screen analysis, in that it is a relatively simple estimation technique providing conservative estimates of the air quality impact of a specific source or source category. CISP performance in evaluating maximum ground-level concentrations has been compared with that of the U.S.EPA Regulatory PTPLU2 Gaussian model (Tirabassi and Rizza [26]). The MAOC model MAOC is a model for elevated point sources in complex-terrain (Tirabassi and Rizza [27]). The simulation of terrain-induced distorsion of flow streamlines is accounted for by modifying the effective plume height. That is: h' = Hi + A h where Hi is the hill height at the receptor considered and A is an empirical factor. As the plume passes over the mountains and streamlines converge, that effective stack height decreases from h to a new value h'. In stable conditions, the plume may not have enough kinetic energy to climb the mountain. In this case, a critical height is defined so that, if h is less than the critical height, the plume will impinge on the mountain, otherwise it passes over the crest of the hill. Model performances have been evaluated using

6 528 Computer Simulation wind tunnel measurements of pollutant concentrations from elevated source in the presence of rough hills and a neutrally stable flow and have been compared with that of COMPLEX 1, the Gaussian model proposed by the U.S.EPA. CONCLUSIONS In practice most of the estimates of dispersion of gas and particulate in the boundary layer are based on the Gaussian approach. A basic assumption for the application of this approach is that the material is dispersed by homogenous turbulence. However, due to the presence of the ground, turbulence is usually not homogeneous in the vertical direction. In this paper we have presented some advanced practical models that use solutions of the advection-diffusion equation that accept wind and eddy diffusivity profiles that are power functions of the height, that is, based on more realistic assumptions. The performance of these models has been assessed with success in cases of both ground level sources as well as elevated sources. REFERENCES 1. Weil J.C. 'Updating applied diffusion models' J. Clim. AppL Meteor., Vol. 24, pp , Holtslag A.A.M. and van Ulden A.P. 'A simple scheme for daytime estimation of surface fluxes from routine weather data' J. dim. AppL Mfffor, Vol. 22, pp , Van Ulden A.P. and Holstlag A.A.M. 'Estimation of atmospheric boundary layer parameters for diffusion applications' J. dim. AppL Mf%w., Vol. 24, pp , Trombetti F.,Tagliazucca M., Tampieri F. and Tirabassi T. 'Evaluation of similarity scales in the stratified surface layer using wind speed and temperature gradient' Atmos. Environ., Vol 20, pp , Beljaars A.C.M. and Holtslag A,A.M. 'A software library for the calculation of surface fluxes over land and sea' Environ. Soft., Vol. 5, pp , Van Ulden A.P. 'Simple estimates for vertical diffusion from sources near the ground' Atmos. Environ., Vol. 12, pp , Berkowicz R.R., H.R. Olesen and U. Torp The Danish Gaussian air pollution model (OML): description, test and sensivity analysis in view of regulatory applications' Proceeding of NATO-CCMS 16th Int. Meeting on Air Poll. Modelling and Its Applications, C. De Wispelaere, F.A.

7 Computer Simulation 529 Schiermeier and N.V: Gillani Ed. (Plenum Press, New York, N.Y (USA), pp , Tirabassi T. 'Analytical air pollution advection and diffusion models' Wafer, Xz'r<W&%'/ /W/., Vol. 47, pp , Hanna S.R. and Paine R.J. 'Hybrid plume dispersion model (HPDM) development and evaluation' J. Appl Meteor., Vol.28, pp Carruthers D.J., Holroyd R.J., Hunt J.C.R., Weng W.S., Robins A.G., Apsley D.D., Smith F.B., Thomson D.J. and Hudson B. 'UK atmospheric dispersion modelling system' In Air Pollution Modeling and its Application IX (ed. van Dop H. and Kallos G.) pp , Proceeding Modeling and its Application, Greta, Greece, Sept 29 - Oct. 4, Plenum Press, New York, Roberts O.F.T. The theoretical scattering of smoke in a turbulent atmosphere' P/w. #oy. &;c., Vol. 104, pp , Rounds W. 'Solutions of the two -dimensional diffusion equation' Trans. )\y. CMo/?, Vol. 36, pp , Smith F.B. The diffusion of smoke from a continuous elevated point source into a turbulent atmosphere' J. Fluid Mech Vol 2 pp Smith F.B. Convection-diffusion processes below a stable layer. Meteorological Research Committee, N and 1073, London. IS.Scriven R.A. and Fisher B.A. The long range transport of airborne material and its removal by deposition and washout-ii. The effect of turbulent diffusion' Amos. Environ., Vol. 9, pp. 59-, Yeh G.T. and Huang C.H. Three-dimensional air pollutant modeling in the lower atmosphere' Bound. Layer Meteor., Vol. 9, pp , Berlyand M.Y. Contemporary problems of atmospheric diffusion and pollution of the atmosphere Translated version by NERC, US EPA Raleigh, NC, U.S.A., 1975, 18. Demuth C. 'A contribution to the analytical steady solution of the diffusion equation for line sources' Atmos. Environ., Vol 12 pp , 1978.

8 530 Computer Simulation 19. Nieuwsadt F.T.M. 'An analytical solution of the time-dependent, onedimensional diffusion equation in the atmospheric boundary layer' Armas. Environ., Vol. 14, pp , Nieuwstadt F.T.M. and de Haan B.J. 'An analytical solution of onedimensional diffusion equation in a non-stationary boundary layer with an application to inversion rise fumigation' Atmos. Environ., Vol. 15, pp , Catalano G.D. 'An analytical solution to the turbulent diffusion equation with mean vertical wind' Proceeding of 16t Southeastern Sem. Thermal. ScL, pp , Miami, Fl, U.S.A., April, Huang C.H. 'A theory of dispersion in turbulent shear flow' Atmos. %., Vol. 13, pp , Tagliazucca M., Nanni T. and Tirabassi T. 'An analytical dispersion model for sources in the surface layer' Nuovo Cimento, Vol. 8C, pp , Tirabassi T., Tagliazucca M. and Zannetti P. 'KAPPA-G, a non-gaussian plume dispersion model: description and evaluation against tracer measurements' JAPCA, vol. 36, pp , Tirabssi T., Tagliazucca M. and Paggi P. 'A climatological model of dispersion in an inhomogeneous boundary layer' Atmos. Environ., Vol. 23, pp , Tirabassi T. and Rizza U. 'An analytical model for a screen evaluation of the environmental impact from a single point source' Nuovo Cimento, Vol. 15C, pp , Tirabassi T. and Rizza U. 'An air pollution model for complex terrain' in Air Pollution (Ed. Zannetti P., Brebbia C.A., Garcia Gardea J.E. and Ayala Milian G.), pp , Proceeding of Air Pollution conference, Monterrey, Mexico, Computational Mechanics Pub. (Southampton) and Elsevier (Amsterdam), 1993.

Abstract. 1. Introduction

Abstract. 1. Introduction Evaluation of air pollution deposition in Venice lagoon T. Tirabassi,* P. Martino,* G. Catenacci,' C. Cavicchioli' "Institute offisbatofc.n.r., via Gobetti 101, Bologna, Italy *Institute oflsiata ofc.n.r.,