Jul 24, 1997 - Richard S Collett and Kehinde Oduyemi, School of Construction and Environment, .... Both schools of lutant across the domain. If a best ...... cerns the suitable evaluation of the Lagrangian auto- ..... New Jersey: Prentice Hall.
Meteorol. Appl. 4, 235–246 (1997)
Air quality modelling: a technical review of mathematical approaches Richard S Collett and Kehinde Oduyemi, School of Construction and Environment, University of Abertay Dundee, Bell Street, Dundee, DD1 1HG, Scotland, UK
A short review paper is presented for the subject area of air quality modelling. The paper is geared towards equipping new researchers and workers with a basic appreciation for the technical aspects of their field, providing a staging point for further investigation, and highlighting useful source materials. The paper is introduced through a discussion of the philosophical and practical implications regarding mathematical air quality modelling. A critique of relevant mathematical modelling techniques is presented and includes a treatment of Box, Gaussian, Eulerian, Lagrangian and Particle modelling approaches. Conclusions on the future of mathematical air quality modelling are drawn.
1. Introduction Before delving headlong into a technical review of modelling methodology, the authors would like to furnish a few definitions for the terms ‘air quality’ and ‘air quality modelling’ Air quality can be defined as a ‘measure of the degree of ambient atmospheric pollution, relative to the potential to inflict harm on the environment’. The concept of threat to public health is fundamental to this description and represents the driving motive behind current air quality research. The potential for deterioration and damage to both public health and the environment, through poor air quality, has been recognised at a legislative level, culminating in the establishment within the United Kingdom of the Environmental Protection Act 1991 (Lane, 1995), and the Environment Act 1995 (Tromans, 1995). Air pollution modelling can be viewed as the attempt to predict or simulate, by physical or numerical means, the ambient concentration of criteria pollutants found within the atmosphere of a domain. The principal application of air pollution modelling is to investigate air quality scenarios so that the associated environmental impact on a selected area can be predicted and quantified. Air quality modelling comprises two very distinct approaches, (Pielke, 1984; Zannetti, 1990). The first approach is known as physical modelling, and attempts to reproduce observed meteorological and air quality conditions through subjecting scaled replicas of a domain to a series of controlled flow regimes with varying physical input parameters, e.g. air temperature, velocity, etc. Modelling of this nature is typically performed in a wind or water tunnel. A scale model of a
specific structure, i.e. building, street, terrain, etc., is placed in the tunnel and subjected to particular flow conditions. Tracer elements representing airborne pollutants are then released upwind/stream of the structure and its transport downwind/stream monitored using appropriate sensing instrumentation. The second approach is classified as mathematical or computational modelling, and attempts to reproduce or predict air quality scenarios, by the intimation of mathematical and physical relationships. When these relationships become too tedious or complex to be used analytically, they are often expressed in algorithmic form and solved using computers. It is this second approach which is the intended focus of this paper. Numerical modelling of air quality scenarios has been identified as being important in several ways (Seinfield, 1986; Milford & Russell., 1993). These are: O To aid in the evaluation of source–receptor relationships so that responsibility for specific impacts can be apportioned. O To aid in project planning, site evaluation and/or environmental impact of present/future sources. O To enable the evaluation of existing sources in relation to compliance with legislation. O To permit the evaluation of proposed abatement and control strategies, in relation to short and/or long term issues. O To permit the assessment of episodic tactics and disaster aversion strategies O To optimise emission inventories and operating conditions while ensuring compliance with legislative controls. O To forecast in real time concentrations of accidental releases. Wyngaard (from Nieuwstadt & Dop, 1982) established what has now become a widely accepted view, that
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R S Collett and K Oduyemi a complete treatment of the physical dynamics of the atmosphere is computationally impossible and will be in the foreseeable future. This conclusion was based on two factors; the inability of current computational technology to provide a complete solution; and a general lack of comprehension regarding the complexities of atmospheric systems and related phenomena (i.e. deposition, precipitation, etc.). Mathematical modelling presents itself within this argument as a half way house, often affecting a partial treatment which can be accurate enough to satisfy the empirical requirements of a given modelling application. For this reason, mathematical modelling is viewed as one of the best mediums/tools through which scientists and environmental engineers can investigate the dynamics of the atmosphere. Historically two equally valid mathematical approaches have been employed in describing air quality dynamics. These are: O The empirical/statistical approach attempts to express the entire spectrum of system behaviour using statistical distributions and probability theory. O The deterministic approach seeks to employ physical laws and simplifying assumptions, to construct a dynamic model based on partial differential equations. Of the two approaches, deterministic modelling is commonly favoured over a statistical approach as it deals with general case solutions and can be easily applied to decision making processes. Both schools of thought have spawned a variety of models, each with their relative merits and applications. The following sections present a short critique on those modelling techniques commonly applied within the field of air quality. The authors have exercised brevity in technical matter as the primary intention was to provide the researcher with an initial staging point from which to investigate further. The review assumes that the reader is aware or familiar with the structure and meteorology of the Earth’s boundary layer (consult Pielke, 1984, Panofsky & Dutton, 1984 or Brown, 1991) for further source material). The review additionally assumes that the reader has a basic regard for modelling and the importance of such concepts as input data, model sensitivity, time scales, etc.
2. Simple air quality models 2.1. Box models The simplest approach to estimating pollutant concentrations over a given domain, is to implement a single box model (Lettau, 1970). The model likens the airshed of a domain to a rectangular box, inside which the mass of the pollutant is fully conserved. The box is orientated in such a way that the directional compon-
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ent of the wind velocity is both perpendicular and incident to one face of the box. The model also assumes that the incoming pollution is instantaneously mixed with the surrounding air, creating a homogeneous concentration throughout the airshed. The mass conservation constraint permits the construction of a mass balance equation of the form: dCV = QA + uC WH − uCWH in dt
(1)
where V is the volume described by the box, C is the homogeneous species concentration within the airshed, Cin is the species concentration entering the airshed, Q is the emission rate per unit area of sources within the box, u is the average wind speed normal to the box, A is the horizontal area of the box (L×W ), W is the width of the box and H is the mixing depth. Integrating equation (1) provides a steady state estimation of species concentration (Venkatram, 1978), assuming the dynamics of the mixing depth to be quasi-stationary and the source emissions to be constant. In the case of reactive species, the chemical reaction dynamics can be incorporated into the mass balance equation as could wet and dry deposition effects. Work presented by Jensen & Petersen (1979), demonstrated a good agreement between observed daily average pollution levels and predictions generated by a single box model. The concentration forecast was shown to lag the observed data by approximately 2 hours in predicting the transient behaviour of the pollutant across the domain. If a best fit time-shift correction is applied to the pollutant concentration forecast, then the magnitude of the predicted pollutant concentrations lies within a 15% error margin with respect to the observed data trend. Topc¸u et al. (1993) employed a single box model in estimating the daily polluting concentration over the Erzurum city in Turkey. Regression coefficients ranging from 87.5% to 97% were obtained when the predicted levels of sulphur dioxide were compared with the observed daily averages. Such high regression factors are typically rare in single box modelling applications though they do serve to demonstrate the possible accuracy of the approach. The application of the single box model is confined to those problems whose objective is the estimation of average pollution levels, across a specific airshed. The size of the airshed is typically large, given that the founding assumptions of homogeneity and instantaneous mixing break down at smaller scales of interest. It is worth noting that by allowing the mixing depth (H) to be a function of the downwind distance x, socalled box models can be employed over smaller scales. In addition, the single box model is incapable of imparting any spatial information regarding the
Review of air quality modelling dispersive nature of a pollutant. This precludes the box model approach from a significant proportion of air quality modelling applications. The method is computationally fast and is capable of providing satisfactory predictions, particularly for scenarios where detailed information on the domain and meteorological conditions is unavailable.
2.2. Gaussian models The Gaussian model forms the basis for the majority of air pollution models, and is the most well known and documented approach. The model presupposes that the dispersion associated with the polluting species can be described by a modified Gaussian or ‘normal’ distribution curve. A three-dimensional axis system is employed to provide a downwind, crosswind and vertical resolution. The species concentration is defined as being proportional to the emission rate of the source, diluted by the wind velocity at the source of emission. The dispersion behaviour of a pollutant is determined by the standard deviations associated with the Gaussian distribution function. These standard deviations are typically functions of atmospheric stability, localised turbulence and distance downwind from the source. The model is usually aligned so that the downwind axis corresponds to the direction of the prevailing wind. The model equation is derived from basic considerations of the diffusion of gaseous matter in threedimensional space.
C=
Q 2πuσ y σ z
− y2 −( h − z )2 exp 2 exp 2 2σ z 2σ y
−( h + z )2 + rG exp 2 2σ z
( 2)
where C is the species concentration at a location (x, y, z), Q is the source emission rate, u is the average wind speed normal to the box, σy is the standard deviation of the horizontal crosswind distribution of the plume concentration and is a function of the downwind distance x, σz is the standard deviation of the vertical crosswind distribution of the plume concentration and is a function of the downwind distance x, h is the effective source height to which the plume has risen, rG is the ground reflection coefficient where (0 # rG # 1), y is the crosswind distance and z is the receptor height above ground. The effective source height (h) or plume rise is the height to which an emission will initially rise as a result of thermal buoyancy and vertical momentum. The upward movement of the plume is retarded on mixing with ambient air reaching an equilibrium point when the internal energy of the plume is equal to that of the
surrounding atmosphere. A review of various semiempirical methods for the estimation of plume rise can be found in Zannetti (1990). Several assumptions are implied in the derivation of equation (2), including: O The emission characteristics of the source are uniform and time invariant. O The meteorological conditions within the domain of interest are homogeneous and time invariant. O The topography of the domain is weak (i.e. flat), so as not to affect pollutant dispersion. O The plume is symmetrical with a straight line trajectory. O The emission originates from a point source where the pollutants are both conservative and passive. O A proportion of the plume is reflected back from the ground plane. O No chemical reactions or deposition effects are accounted for. These assumptions are not borne out in reality, as emissions and meteorological conditions can be highly variable and non-linear by nature, i.e. a doubling in emissions may not incur a corresponding change in the downwind concentration of pollutants, or for that matter in the concentration of any secondary pollutants. This factor is particularly true in conditions of increasing turbulence, in which distortions in the trajectory and dispersion of the plume can mask out any discernible relationship between the downwind pollutant concentration and its source. Pollutants are also seldom passive and emissions are typically dependent on plant operation. In light of such limitations, the Gaussian model can only be considered workable when such factors are static enough to be regarded as homogeneous. The predictive capabilities of the model are documented by the Canadian Standards Association, CSA (1991). For distances less than about 1 km under ideal conditions of uniform flat terrain, steady meteorology etc., the prediction of the maximum downwind ground level time integrated concentration is only accurate to within 20% for a ground level release and to within 40% for an elevated release. The accuracy of the model within ± 2σy off the centreline is estimated to be within a factor of two. When the Gaussian model is applied to reasonable conditions of meteorology, terrain, and distances up to 10 km, the scatter of the estimate of the average time integrated concentration is expected to increase and the accuracy is then estimated to be within a factor of two. Under certain conditions the equation (2) can be reduced to a variety of simplified forms. Consult
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R S Collett and K Oduyemi Masters (1991) and Boubel et al. (1994) for further reading.
curve represents the value of the dispersion coefficient at a specific stability class.
The behaviour of the Gaussian model depends heavily upon the correct calculation of the dispersion coefficients, σy and σz. Various approaches currently exist and a summary of the methods for the estimation of σy and σzcan be found in Zannetti (1990) and Boubel et al. (1994). The most common method of calculation is the semi-empirical method.
A formula method of estimating the dispersion coefficient was proposed by Martin (1976). Again, an exponent law was employed in obtaining the coefficient values for downwind distances, over short and long scales of interest (see equation (4) and Table 2).
σ y = a x 0.894
σ z = c xd + f
( 4)
Similar formulations have been proposed by several workers (e.g. Irwin, 1980; Green et al., 1980; Briggs, 1973). The latter is of particular note as estimates of the Pasquill dispersion coefficients over urban and rural domains are provided.
(a) Semi-empirical calculations of σy and σz In the likelihood that specific information regarding the variation of the wind is unattainable, a second approach, utilising routine meteorological observations to provide a semi-empirical estimation of the dispersion coefficients, can be considered. Work by Smith (1951) and Singer & Smith (1953) led to a method for estimating the stability of the atmosphere using a wind direction trace obtained over 1 h. Five stability classes were proposed from the empirical data collected. The dispersion coefficients were calculated using equation (3), where the relevant coefficients, referenced to the appropriate stability class, can be seen in Table 1. Note both σy and σz have units of m where x is in km.
(b) Advanced modifications and applications
In order to overcome several of the limitations inherent within ‘simple’ Gaussian models, many workers have suggested additional modifications. Green et al. (1980) discussed various analytical extensions of the Gaussian Plume Model, specifically the extension of the model into polar co-ordinate space so that an analytical solution could be developed, utilising wind velocity input data in a wind rose format. This approach σ y = a xb σ z = c xd (3) sought to overcome the dependence of the Gaussian model on a constant velocity input vector, and to proA similar approach was suggested by Pasquill (1961). vide a facility for a 360° scope of prediction. Pasquill advocated the use of fluctuation measurements for dispersion estimates but provided a scheme A segmented approach (Hales et al., 1977) was ‘for use in the likely absence of special measurements developed to account for variations in the source emisof wind structure’. The scheme did not require soph- sion and meteorology. The virtual plume is dissected isticated measurement equipment but utilised basic into a number of segments whose dynamics are funcvalues of wind speed, insolation and cloudiness in the tions of the emission rate and the prevailing meteoroestimation of atmospheric stability. However, it was logical conditions, and are oriented according to the Gifford & Sklarew (1961) who restated the Pasquill time varying wind direction. The model suffers from stability classes in terms of σy and σz so as to permit supposition effects, where the error associated with a their use in the Gaussian plume equations. A compar- pollutant concentration close to the ends of the segison between the gustiness classes and the Pasquill ment, varies proportionally with the angle between the stability classes are given in Gifford (1976). Turner consecutive segments. (1970) and Gifford (1976) graphically related the Pasquill dispersion parameters to the downwind distance More recently, van Jaarsveld et al. (1993) proposed a from the source. The dispersion coefficient is directly Gaussian ‘operational model for priority substances’ obtained by estimation from the graph, where the to predict the concentration and deposition removal of Table 1. Coefficients and exponents for gustiness stability classes Stability class
a
b
c
d
A B1 B2 C
— 0.36 0.40 0.32
— 0.86 0.91 0.78
— 0.33 0.41 0.22
— 0.86 0.91 0.78
D
0.31
0.71
0.06
0.71
η = fluctuation in wind direction. From Gifford (1976) and Singer & Smith (1953).
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Classification Highly unstable, η>> 90° Unstable, 45°
