**1. Introduction**

The scientific community has been driven to act urgently due to the increase in ecological disasters worldwide. A key approach to addressing this issue is the development of reliable models that enable quantitative prediction of pollutionrelated phenomena, either through analytical descriptions or simulations using powerful and operational tools. These models, which can be based on simulations or analytical descriptions, have strong quantitative predictive power for understanding pollution-related phenomena. The atmosphere is considered the primary means of dispersing pollutants in the environment, which can come from

industrial sources or accidental events, leading to progressive contamination of the ecosystem, fauna, flora, and all populations. Therefore, accurately evaluating how pollutants move through the atmosphere in the boundary layer is vital to preserve, protect, and restore the integrity of ecosystems. To accomplish this, it is essential to create atmospheric dispersion models tailored to the parameters and weather conditions specific to the region being studied, using parameters, weather conditions, and local topographic information. These models should produce realistic outcomes on environmental consequences, helping to minimize the negative impact of potential disasters like forest fires. Furthermore, implementing simulations of models based on specific cases can be beneficial in establishing particular emission limits for industrial sites, limiting the release of pollutants into the air.

The aim of our work is to propose a closed-form analytical solution for threedimensional advection-dispersion transport problems in finite, multilayered media using rigorous mathematical tools. We ensure that wind velocity profiles and vertical diffusivity coefficients take average values in each sub-layer. We have made innovative contributions by using a governing function to generate the eigenvalues associated with our problem and overcoming the common difficulty of missing some of the eigenvalues when they are calculated by developing a transcendental equation for each layer. Finally, our approach has enabled us to obtain a closed-form analytical solution that differs from previous solutions that required the determination of integral coefficients. This method could have important implications in many scientific and technical fields for solving complex advection-dispersion transport problems in finite, multilayered media.

In recent years, there has been an increasing emphasis on creating new analytical approaches that can be used for a range of wind speeds and turbulent diffusivity coefficients. For the most part, these approaches involve the projection of the solution onto a basis of orthogonal polynomials, such as the GILTT technique [1–8]. However, an important drawback of this method is that it requires a large number of eigenvalues to ensure convergence, which can reach up to 250 eigenvalues. In contrast, our new solution is able to provide better results by using a much smaller number of eigenvalues, specifically between 10 and 15, to ensure convergence. This is a significant improvement over existing approaches, making our method more efficient and practical for analytical applications.

Our model was developed using the Fourier transform and separation of variables technique, which resulted in the Sturm-Liouville problem. In order to understand the factors influencing pollutant dispersion, we considered the following parameters: (*i*) the wind speed profile of Deaves and Harris [9], (*ii*) the vertical turbulent diffusivity coefficient, which is considered as an explicit function of the downwind distance and vertical height under convective conditions, as described by Mooney and Wilson [10] and Degrazia et al. [11], and (*iii*) the lateral eddy diffusivity coefficient, which also depends on the downwind distance and vertical height, as described in the work of Huang [12] and Brown et al. [13]. We first proceed to the exposition of the general formulation of the equation that governs the dispersion of pollutants in the atmospheric boundary layer, with a comprehensive presentation in Section 2. We then turn to the explicit solution in Section 3, while exploring the model parameterizations in Section 4. We provide an in-depth discussion of our numerical results in Section 5. Finally, the last section proposes our conclusion, a synthesis of our investigation.

*A Novel Development in Three-Dimensional Analytical Solutions for Air Pollution… DOI: http://dx.doi.org/10.5772/intechopen.112225*

### **2. Presentation of the general problem**

The movement of pollutants within the planetary boundary layer (PBL) is characterized by turbulent dispersion, which is governed by a mathematical equation known as the advection-diffusion equation. This equation considers the combined influence of two primary mechanisms, advection and diffusion, which respectively refer to the transport of pollutants by wind and the spreading of pollutants due to turbulence.

By applying the advection-diffusion equation, researchers can gain a more comprehensive understanding of the intricate and dynamic behavior of pollutants within the PBL. This equation serves as a mathematical framework that accounts for various factors that affect pollutant dispersion, such as wind direction and speed, Eddy diffusivities, and atmospheric stability, and pollutant concentration, presented in the following form:

$$\frac{\partial \mathbf{C}}{\partial t} + \nabla . (\mathbf{U\_w} \cdot \mathbf{C}) = \nabla . (D \cdot \nabla \mathbf{C}) + \mathbf{S}, \tag{1}$$

where **Uw** <sup>¼</sup> ð Þ *<sup>U</sup>*,*V*, *<sup>W</sup> <sup>T</sup>* is the wind speed vector (*m/s*) representing the components *U*, *V*, and *W* in the east-west, north-south and vertical directions, respectively; *D* is the molecular diffusion coefficient; *S* is the source term; and ∇ is the gradient operator.

By use of the time average and fluctuation values, *U* ¼ *u* þ *u*<sup>0</sup> , *V* ¼ *v* þ *v*<sup>0</sup> , *W* ¼ *w* þ *w*<sup>0</sup> and *C* ¼ *c* þ *c*<sup>0</sup> , the wind speed vector **Uw** is expressed as:

$$\mathbf{U\_w} = \overline{\mathbf{U\_w}} + \mathbf{U\_w}', \quad \text{with} \quad \overline{\mathbf{U\_w}} = (u, v, w)^T \quad \text{and} \quad \mathbf{U\_w}' = (u', v', w')^T \tag{2}$$

Applying the Reynolds averaging rules [14] to the vertical mass flow term, denoted as **Uw** *C*, results in the following expression: It turns out that turbulent diffusion can be described with Fick's laws of diffusion as follows [15].

$$
\overline{u'c'} = -K\_{\text{x}} \cdot \frac{\partial \mathcal{E}}{\partial \mathbf{x}}; \quad \overline{v'c'} = -K\_{\text{y}} \cdot \frac{\partial \mathcal{E}}{\partial \mathbf{y}}; \quad \overline{w'c'} = -K\_{\text{x}} \cdot \frac{\partial \mathcal{E}}{\partial \mathbf{z}}.\tag{3}
$$

where *Kx*, *Ky*, and *Kz* are the eddy diffusivities components along *x*, *y*, and *z* directions, respectively.

It should be noted that in a turbulent boundary layer where advection is occurring, *K* will be larger than *D* and eddy diffusion will dominate solute transport. In this case, the molecular diffusion coefficient ∇*:*ð Þ *D* ∇*C* is then to be replaced by an eddy or turbulent diffusivity. The source term could be eliminated from Eq. (1) and should be added to the boundary conditions as a delta function: At the point 0, 0, ð Þ *Hs* , there is a source rejecting the pollutant with a continuous flow *Q*:

$$
\mu \lrcorner c(\mathbf{0}, \mathbf{y}, \mathbf{z}) = Q \lrcorner \delta(\mathbf{y}) \circledast \delta(\mathbf{z} - H\_{\mathfrak{s}}), \tag{4}
$$

where ⊗ is the tensor product of two distributions and *Hs* is the source height. By application of the Reynolds averaging and the divergence operator to Eq. (3), Eq. (1) may be written as follows.

$$\frac{\partial \mathcal{L}}{\partial t} = -\frac{\partial}{\partial \mathbf{x}} \overline{u'c'} - \frac{\partial}{\partial \mathbf{y}} \overline{v'c'} - \frac{\partial}{\partial \mathbf{z}} \overline{w'c'} - \overline{U\_w} \nabla c. \tag{5}$$

In the remainder of this paper, the following assumptions are considered:


These assumptions lead to the steady-state advection-diffusion equation defined as:

$$\begin{cases} u(x)\frac{\partial c}{\partial x} = \frac{\partial}{\partial y} \left( K\_{\mathcal{V}} \frac{\partial c}{\partial y} \right) + \frac{\partial}{\partial x} \left( K\_{x} \frac{\partial c}{\partial x} \right); & (x, y, x) \in [0, L\_{x}[\times]] - L\_{\mathcal{V}}, L\_{\mathcal{I}}[\times] \mathbf{0}, H\_{mix}], \\\\ \text{subject to the boundary conditions}: \begin{cases} u(x) \ c(x, y, z) = \mathbf{Q} \cdot \delta(y) \otimes \delta(x - H\_{l}); & x = 0 \\\\ K\_{\mathcal{V}}(x, z) \frac{\partial c(x, y, z)}{\partial y} \to \mathbf{0}; & |y| \to L\_{\mathcal{V}}, \\\\ K\_{z}(x, z) \frac{\partial c(x, y, z)}{\partial z} \quad = \mathbf{0}; & z \in \{0, H\_{mix}\}. \end{cases} \end{cases} \tag{6}$$

where *z*<sup>0</sup> is the surface roughness length and *Hmix* is the PBL height. We consider that the eddy diffusivities have the following separable formulations:

$$K\_{\mathcal{Y}}(\mathbf{x}, \mathbf{z}) = \zeta\_{\mathcal{Y}}(\mathbf{x}) \text{ } \mathfrak{u}(\mathbf{z}), \tag{7}$$

$$K\_x(\varkappa, z) = \xi(\varkappa) \,\,\varrho\_x(z). \tag{8}$$

We vertically divide the PBL into *H* intervals such that for each one the eddy diffusivity and wind speed assume average values. For *h* ¼ 1, ⋯, *H*,

$$
\mu\_h = \frac{1}{z\_h - z\_{h-1}} \int\_{x\_{h-1}}^{x\_h} \mu(s) ds,\tag{9}
$$

$$
\rho\_{x\_h} = \frac{1}{z\_h - z\_{h-1}} \int\_{x\_{h-1}}^{x\_h} \rho\_x(s) ds. \tag{10}
$$

By use of the formulations of *Ky* and *Kz* given by Eqs. (7) and (8), Eq. (5) is written as:

$$
\mu\_h \cdot \frac{\partial c\_h}{\partial \mathbf{x}} = \zeta\_y(\mathbf{x}) \cdot \mu\_h \cdot \frac{\partial^2 c\_h}{\partial \mathbf{y}^2} + \xi(\mathbf{x}) \cdot \varrho\_{x\_h} \cdot \frac{\partial^2 c\_h}{\partial \mathbf{z}^2},\tag{11}
$$

with *uh* and *φzh* (given by Eqs. (9) and (10)) are constants.

This later Eq. (11) is subject to the first boundary conditions Eq. (6) on the one hand, and on the other hand, the continuity of both the concentration and the flux at the interface level is applied.

*A Novel Development in Three-Dimensional Analytical Solutions for Air Pollution… DOI: http://dx.doi.org/10.5772/intechopen.112225*

$$\begin{cases} \begin{aligned} &\varrho\_{x\_{1}}\frac{\partial c\_{1}(\mathbf{x},\mathbf{y},z\_{0})}{\partial \mathbf{z}} = \mathbf{0}, \\ &\begin{cases} &c\_{h-1}(\mathbf{x},\mathbf{y},z\_{h-1}) = c\_{h}(\mathbf{x},\mathbf{y},z\_{h-1}) \\ & \end{cases} \\ &\begin{aligned} &\varrho\_{z\_{h-1}}\frac{\partial c\_{h-1}(\mathbf{x},\mathbf{y},z\_{h-1})}{\partial \mathbf{z}} = \varrho\_{z\_{h}}\frac{\partial c\_{h}(\mathbf{x},\mathbf{y},z\_{h-1})}{\partial \mathbf{z}} \\ &\varrho\_{z\_{H}}\frac{\partial c\_{H}(\mathbf{x},\mathbf{y},z\_{H})}{\partial \mathbf{z}} = \mathbf{0} \end{aligned} \end{cases}, \ \forall h \in \{2,\cdots,H\}, \\\\ &\begin{aligned} \varrho\_{x\_{H}}\frac{\partial c\_{H}(\mathbf{x},\mathbf{y},z\_{H})}{\partial \mathbf{z}} = \mathbf{0} \end{aligned} \end{cases}$$
