## A indicates the infinite diffusivity inside the ideal box

A Review on Box Modeling of Air Quality Studies Wang ZhizhaoAbstractThe box model is one of the most basic air quality models for dispersion at a ground level. During the 50-year development, the box model has been becoming more precise by incorporating with targeted experimental and mathematical methods and adding representative parameters such as mixing height (MH). This review paper describes the widespread application and explicit advancement of box modeling in the field of air quality estimation and prediction since it is first described in 1968. The development and the newest studies with respect to box modeling on distinctive contaminants (O3, CO, PM2.5, etc.) are presented in this paper corresponding to indoor cooking stove emission, transportation pollution in street canyons, urban air quality and photochemistry study. Keywords: box modeling; air quality; air pollution; street canyon; IAP; photochemistry.1. IntroductionAir quality is defined as a ‘measure of the degree of ambient atmospheric pollution, relative to the of potential to inflict harm on the environment’ 1, which has a substantial potential to public health and environment. To estimating and forecasting air quality, mathematical formulations so-called air quality models that include parameters that affect pollutant concentrations are used. The box model is one of the basic air quality models describing air dispersion, vertical mixing and moisture in time and space. The first so-called fixed box model was brought out in 1968 by Stern 2 and then used by Lettau to simulate air pollution pattern in the urban area 3. The model assumes a perfect mixing within the box and the air velocity is perpendicular to the inlet surface, which indicates the infinite diffusivity inside the ideal box model. In 1973, Kenneth enriched the concept by counting on the geostrophic wind, net heat flux, surface roughness, mixing height of the atmosphere and emission rates 4. The box model thence has been wildly attempted and indicated as an ideally suited for the study of air pollution climatology and environmental impact assessment 56. 1.1. Fixed box modelsThe simplest three-dimensional fixed box model can be described by figure 1 below. Generally, a box model of air quality is a defined system based on the principle of mass conservation. The model receives sources (origin of pollution) and simultaneously removes the contaminants (deposition or emission). The sources could be generated by first-hand emission, ventilating, chemical processes and re-suspension, while ventilating, chemical processes and deposition hinges the remove processes. Figure 1: A basic fixed box model 1 Consider a three-dimensional, rectangular Eulerian coordinate system and the conservation of mass for the air and the pollutants, the solution of the basic fixed box model is given by 2, (1) Where, Ci is the concentration of the pollutant in the ambient, u is the velocity of air (with pollutant), C is the concentration of the pollutant within the box and the exit, ? is the source term, X, Y and zx are the length, height and width of the box in the direction of the airflow respectively. The pollutants are assumed to be completely mixed within a single box which covers the city and extends upward to the mixing height 7. And the air mass is considered as well mixed and concentrations to be uniform throughout. The advantage of the fixed box model is simple meteorology input and detailed chemical reaction schemes, detailed aerosol dynamics treatment. However, following inputs of the initial conditions a fixed-box model simulates the formation of pollutants within the box without providing any information on the local concentrations of the pollutants 8. 1.2. Multi-box models After the basic fixed box model, Kenneth (1973) described a developed box model adding crucial inputs to specify the geostrophic wind, the net heat flux, the surface roughness, the mixing height of the atmosphere and the emission rate of the source. In this model, the ambient air concentration of pollutants is dependent primarily upon the emission rate of pollutants, the surface “aerodynamic roughness”, the net heat flux, the mixing height and the wind speed at the mixing height 47.Comparing with the single layer fix model which has a constant height, the multi-box has a variable mixing height. The figures 2 and 3 below show the physical and mathematical of the muiti-box model, which indicates the difference between fixed box model and muiti-box model clearly. Figure 2: Physical layouts of the multi-box model 4 Figure 3: Mathematical layouts of the multi-box model 4 In figure 3, the mathematical procedure considers a 4 x 4 array of boxes, whose boxes are ?X wide, ?Z high and 1 unit deep. Consider the conservation of mass for a steady flow of a non-reacting species with turbulent diffusion in the vertical direction (with Fick’s Law assumed) is showed as 4, (2)with boundary conditions at,x = 0, C = B (3)z = 0, (4)zmix = 0, (5)Where, u=u(z) is the mean wind speed in the x direction, C=C(x,t) is the ambient concentration of the pollutant species, Kz=Kz(z) is the turbulent eddy diffusivity, B=B(z) is the background concentration of the pollutant species, Q=Q(x) is the emission rate of the pollutant species and z is the mixing height. For multi-box model, the mass-conservation equation is solved for each of the boxes. There is no diffusion between the boxes and the concentration in each cell is uniform 8.1.3. Photochemical box model (PBM)Photochemical model is one of the chemical models to simulate air quality, which describes transformation of directly emitted particles and gases to secondary particles and gases, and also estimates the equilibrium between gas and particles for volatile species 1. A PBM is a combination of box modeling and chemical modeling which has a better performance when applied to further investigate the differences and similarities of the pollutant-precursor photochemical relationships, the atmospheric photochemical reactivity and production. The first PBM for urban air quality study was developed by Jin and Demerjian in 1993 9. The PBM is based on the principle of mass conservation, and it assumes that :• The box volume is well mixed at all times and there is no spatial variations of concentration occur inside the box;• The emission sources for the model are homogeneously distributed across the bottom surface of the box;• The transportation of outside air occurs laterally by advective transport and vertically by the growth in mixed layer height.Under these assumptions, the PBM can be illustrated as figure 4 below, and the chemical species conservation equation becomes 9, (6) Where, Ci is the mean concentration of species i within the PBM domain, U the mean advection speed, Qi is the source emissions flux of species i in the domain, Ri is the rate of production and destruction of species i due to chemical reactions. The PBM has a horizontal extension of 20 km and a vertical extension of the mixed-layer height 9.Figure 4: The illustration of PBM in 1993 9 One of the main purposes of PBM is to assess the photooxidation cycle in the polluted troposphere, which is initiated by free radical processes. These free radicals are predominantly photochemical in origin, which initiate and propagate a series of chemical reactions 10. The photooxidation cycle is given in Figure 5. Figure 5: The illustration of the atmospheric photooxidation cycle 10 The model has a distinct horizontal domain of the city size and a vertical dimension defined by the mixed-layer height. Besides, it calculates air pollutant concentration based on horizontal advection, vertical entrainment, source emissions and chemical reactions. The simulated pollutant concentrations using both chemical mechanisms are in very good agreement with available observations for CO, NO, NO2 and O3 10. Furthermore, Shon et al. (2005) employed a photochemical box model (PCBM) to estimate reactive gaseous mercury (RGM) concentrations in the urban atmospheric boundary layer (ABL), while the model used a mass balance approach to calculate the pollutant concentrations by considering various processes such as the emission flux into the ABL,chemical reactions and dry deposition, and mass transfer between gas and aqueous phases 11. 1.4. other box modelsBesides the box models introduce in the above, B. Tsuang and J. Chao derived a circuit box model for calculating pollutant concentration from an area source. This model includes various loss, production and transfer terms and can be used by grid modellers seeking to build a more sophisticated description of the to the planetary boundary layer (PBL). They also apply the model to demonstrate that air quality deteriorate under the persistent stable atmosphere with low wind speed and rainless conditions 12.Middleton (1998) forecasted the concentrations of NO2 in urban areas by developing a box model called Boxurb which can use synoptic observations of themeteorology and numerical forecasts of wind and cloud 13. After that, a box model was applied to estimate the contributions of different economic activities to the air pollutant concentrations, while the model explicitly included the seasonal behavior of meteorological variables and considered various processes such as emission, advection, dry and wet deposition, and chemical transformation within the atmospheric mixing height 14Lin et al. (2004) proposed a Lagrangian box model to locate the influential pollution sources and estimate their contributions to pollutant concentrations observed at a receptor site in southern Taiwan by taking into account the effects of source emissions, atmospheric dilution, and chemical transformation and deposition 15. In 2006, Shuiyuan Cheng et al. developed a Gaussian-box modeling approach due to multiple point- and area-source emissions, which improved upon the conventional box models by allowing consideration of more details in spatial variations of emission sources and meteorological conditions 16. 2. ResultsTo apply box models with simulating pollutants changing in real time, the verification and performance of the box model need to be improved during case studies. The accuracy and sensitivity of the box model need to be testified as well. Studies show that the box modeling with targeted methods has a high performance when simulates the transportation pollution in a street canyon, the indoor cookstove emission capabilities, and ozone pollution. 2.1. Multi-box models in a deep street canyonExtensive studies have been conducted to box modeling applying on the urban traffic intersection since 2007. A street canyon is a relatively narrow street in between buildings that line up continuously along both sides, which is also a typical urban configuration with surrounding buildings along the street 17. The typical two kinds of air flow (wake interference flow and skimming flow) in street canyons can be described as the figure 6 below. Figure 6: Two flow regimes of street canyons 17 Sharad Gokhale and Suresh Pandian (2007) presented the development and application of a semi-empirical model based on the box approach for the predictions of hourly CO concentrations from the traffic flow pattern which is heterogeneous in nature and a sub-tropical meteorology. In their study, the model was modified to account for the road wind angles and stability-dependent vehicle wake factors 18.In 2011 and 2012, Murena et al. proposed a distinct two-box model to simulate mass transfer inside deep street canyons and between them and atmospheric flow above based on a Computational Fluid Dynamic (CFD) simulation study on ideal deep street canyons 1920. The model described on figure 7 and the equations of a two-box model for a deep street canyons are given by 19, (6) (7) Where, Hb and Hu are respectively the height of the bottom and upper boxes, cb and cu are the CO concentrations in the bottom and upper boxes, ca is the air concentration above the roofs, fe is a weighted average CO emission factor, Qv is the number of vehicles per hour , and ubu is the mass transfer velocity between the bottom and upper boxes.Figure 7: box models for deal deep street canyons 19 The two box model defines an overall mass transfer coefficient that allows the evaluation of the concentration at the pedestrian level, which is of particular interest for environmental and health impact assessment studies. The application of this two-box model in distinct air contaminants of traffic pollution is then studied by Fabio (2012, CO) in Naples 20 and H. Yun et al. (2017, NOx) in Hong Kong 21. Moreover, Zhong J et al. developed a coupled two-box model approximation based on previous work, which highlights the limitation of the assumption of homogeneity in single box models for street canyon simulation, and the inherent uncertainties that must be borne in mind to appropriately interpret such model output (in particular, that a single-box treatment will systematically underestimate NO2 as experienced at street level) 22. 2.2. Monte Carlo single-box model of indoor air pollutionThe indoor air pollution (IAP) caused by solid fuel cookstoves emission is still a problem in the developing world. The resulting exposures have been estimated to cause 3 – 4% of the global burden of disease 23. To evaluate the performance of different kinds of cookstoves, an advanced box model so-called Monte Carlo single-box model has been developed by M. Johnson et al. In 2011 24.The Monte Carlo single-box model is used to predict kitchen concentrations of air pollutants given emission performance data for various stove/fuel combinations and information about typical cooking needs and kitchen characteristics. For the model, Indoor air pollutant concentrations were modeled assuming a well-mixed room with single constant emission source. The model assumes instantaneous mixing with zero backflows to the room, that removal of the pollutant from the air is dominated by ventilation, and competing for loss mechanisms are negligible. And the model solution is showed as 24, (8) Where, Ct is the concentration of pollutant at time t, G is the emission rate, ? is the first order loss rate (nominal air exchange rate), V is the kitchen volume, t is for time, C 0 is the concentration from preceding time unit, and f is the fraction of emissions that enters the kitchen environment.This capacity can be a useful approach for preliminary, cost-effective evaluation of a stove’s potential IAP impacts, as well as linking health-based air quality guidelines to stove performance standards. 2.3. Surface O3 photochemistryOzone pollution is a complex phenomenon since it involves precursor emissions, photochemical formation and dynamic transport on different scales 25. To estimating ozone pollution in a certain area, a photochemical box model incorporating the Master Chemical Mechanism (PBM-MCM) was applied in terms of O3 -precursor relationship, atmospheric photochemical reactivity and O3 production at a semirural site and an urban site in Hong Kong 252627 and an island site (Wanshan Island, WSI) over the South China Sea (SCS) 28. It was observed that mixing ratios of O3 and its precursors (such as volatile organic compounds (VOCs), nitrogen oxides and carbon monoxide (CO) showed significant differences on non-episode days and episode days. This box model considers chemical reaction and other modules such as photolysis rate, dry deposition and the boundary layer height. Compared to the previous study, this study extends the modeling to further investigate the variations of HOx and its sources and sinks, the contributions of different pathways to O3 production and destruction, and the O3 production efficiency related to OH regeneration 26. 3. ConclusionsThe box model is one of the most widespread modeling approaches in air quality studies. It has been approximate 50 years since it first was described, which is developed with various chemical or mathematical methods. The intention of box modeling is to evaluate and predict the urban air quality, to analyze the relationship between pollution emission and air quality, to provide an improved simulation tool for developing air quality strategy.In this review, the three typical box models (fixed, multi-box and PBM) are introduced, then following three distinctive study fields of the application of box modeling. Nowadays, the studies of box models are focus on the multiple fields such as cookstoves performance testing, dispersion of air pollutants in deep street canyon and ozone photochemistry.The two crucial properties of box modeling are accuracy and sensitivity. In general, due to the simplicity of the model, researchers needs to experiment and advance it with multiple testing and combining/comparing with another modeling approaches. Also, the simulation results of advanced box models could be not universal in another cases. Researchers needs to specifically examine the input and output of box models.For the further improvement of box modeling, there are some approaches could be studied : • Comparing and combining the box model with other air quality models such as the Lagrangian trajectory model and 3D Eulerian grid models. Multiple modeling analysis could increase the precision of simulation to some extent. • Exploring the application and development of box modeling in other fields such as industrial plants and complicate street area. The health of workers in industrial plants is still a burning issue. 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