The emission of pollutants into the atmosphere interferes with air chemical composition and can cause both environmental and human health damage [1] [2] [3]. Several sources are responsible for emitting such pollutants and can be classified as natural, such as natural fires and volcanic emissions, or anthropogenic like burning fossil fuels in factories [4] [5] [6]. Understanding how emissions from these sources occur, especially anthropogenic ones, we try to minimize these damages [7] [8] [9]. Therefore, many researchers have been developing their studies using mathematical modeling [7] [10] [11] [12]. Still, several methods have been used to evaluate pollutant dispersion, considering geometries with point sources like chimney or obstacles [13] [14] [15].

In this context, this work will be conducted in simulations that describe the movement and dispersion of pollutants using mathematical modeling. The numerical solution is designated only for a passive scalar transport, the concentration is solved for a given stationary velocity field, and it can deal with only the constant diffusion coefficient, so that it cannot take into account the turbulent diffusion caused by turbulent flows [15].

The computational mesh that represents the physical domain in the tests was modeled, followed by the numerical model described by the Navier-Stokes equations [16] [17] [18] [19] [20] and the pollutant transport equation [21] [22] [23].

The solutions precision obtained by the simulations of fluid dispersion relies on how well computational mesh represents the studied geometry. For problems with complex geometries, it is convenient to use the curvilinear coordinate system instead of cartesian coordinates, due to the fact that cartesian coordinates lead to a poor boundary fit, since the physical domain doesn’t always coincide with the mesh domain [18] [24] [25] [26]. Another reason that leads to the use of the curvilinear coordinate system, in the discretization of the computational mesh, refers to less difficulty in programming computational codes to solve complex problems and the facility in developing generic methodologies [27].

Despite the advantages of using the curvilinear coordinates for the construction of meshes, it is still possible to obtain low-quality elements. A way to circumvent the quality of the mesh elements, without necessarily improving the refinement, because it would increase the number of nodes in the mesh and, consequently, the computational costs with memory and execution time, refers to the use of the multiblock technique [18] [26] [27] [28] [29]. Applying this technique, the domain is divided into parts, called block or sub-grids, and the equations are solved in each block.

With independent block separation, it is possible to increase mesh refinement at important areas or decrease it in less relevant areas, allowing the generation of structured grids to keep simulations costs balanced. Illustrating a comparison between meshes generated by curvilinear coordinates, consider a single block, Figure 1(a), and multiblocks (5 blocks), Figure 1(b).

For the geometry, Figure 1, the meshes were generated with the same number of partitions. Due to the rectangular form of the geometry and the block division structure, the mesh in curvilinear coordinates resulted in square elements.

Therefore, through the multi-block technique, the quality of the mesh elements can be improved.

Hence, this work proposes a way of modeling two-dimensional meshes using elliptic equations, generating meshes in the curvilinear coordinate system coinciding with the problem geometry. Still, for a better representation of the problem domain, was used the multiblock technique [26] [27] [28] [29]. The velocity field was obtained by simulating the Navier-Stokes equations and then the transport equation is solved, modeling the pollutant dispersion in the mesh. The discretizations are obtained by applying the finite difference method and using curvilinear coordinates. In the Navier-Stokes equations, it is applied the first order upwind scheme in the convective term [17] [26], central differences in the second order terms and progressive differences for the temporal term [18]. In the pollutant transport equation, it uses central differences in terms of first and second order and progressive differences for the temporal term.

To generate the meshes, the blocks were represented using the notation involving the cardinal points, where the W side represents the west direction, E, east, N, north and S, south direction. Then, to generate the blocks, parameter with the number of elements in ξ and η directions and their coordinates points to each side were passed.

The boundary conditions are important for a fluid dispersion numerical solution, since they define the simulation limits and help to guarantee the method stability and convergence [18]. Besides, correctly define the boundaries of the block is extremely important. The implemented algorithm recognizes the boundary conditions type of the four sides of each mesh block.

To solve the Navier-Stokes equations were used these boundary conditions:

• CNEI: no-slip and impermeability condition of the fluid, indicates the wall fluid’s velocity is null and there isn’t flow through the wall as well;

• CIPR: prescript input condition, indicates that at the domain border it has non-zero velocity;

• CECO: continuous fluid flow condition, indicates the domain fluid output;

• ADJA: adjacency blocks condition, indicates the data transferring between adjacent blocks.

The boundary condition to solve the transport equation can be:

• CIPRCONC: prescript concentrations condition, used at the domain border, indicates the concentration that enters into the domain across the border;

• CCONCCONC: continuous concentration condition, indicates the domain concentration output;

• ADJACONC: used to indicate adjacency between blocks.

A concern while using multiblocks technique is to maintain simulation consistency, for this purpose, a treatment occurs in the boundaries that indicate adjacency between blocks [18] [26] [27] [28] [29].

The two bands of previous/posterior elements of neighboring blocks serve as transfer area, with no solution calculation in them. To guarantee simulation continuity, it is required that the adjacent boundaries points have the same coordinate in both neighboring blocks. Figure 2 shows an example of adjacency boundary, illustrating how was accomplished the data transfer between two blocks.

Assuming by hypothesis a laminar flow, newtonian, isothermal and incompressible, in two-dimensional, without the existence of the source terms, then the Navier-Stokes equations described in the curvilinear coordinate system are written by:

∂ ∂ τ ( ρ u J ) ︸ temporal term + ∂ ∂ ξ ( ρ U u ) + ∂ ∂ η ( ρ V u ) ︸ convective term = μ [ ∂ ∂ ξ ( J ( α ∂ u ∂ ξ − β ∂ u ∂ η ) ) + ∂ ∂ η ( J ( γ ∂ u ∂ η − β ∂ u ∂ ξ ) ) ] ︸ diffusive term + [ ∂ p ∂ η ∂ y ∂ ξ − ∂ p ∂ ξ ∂ y ∂ η ] ︸ pressure term (1)

∂ ∂ τ ( ρ v J ) ︸ temporal term + ∂ ∂ ξ ( ρ U v ) + ∂ ∂ η ( ρ V v ) ︸ convective term = μ [ ∂ ∂ ξ ( J ( α ∂ v ∂ ξ − β ∂ v ∂ η ) ) + ∂ ∂ η ( J ( γ ∂ v ∂ η − β ∂ v ∂ ξ ) ) ] ︸ diffusive term + [ ∂ p ∂ η ∂ x ∂ ξ − ∂ p ∂ ξ ∂ x ∂ η ] ︸ pressure term (2)

where the dynamic viscosity μ and the specific mass ρ are constant. For further details of how to describe these equations in this coordinate system see Maliska [18] and Cirilo et al. [26].

The terms U and V, in Equations (1) and (2), are named contravariant components of the velocity vector, these are normal to ξ and η lines respectively, and

defined as U = 1 J ( u ∂ ξ ∂ x + v ∂ ξ ∂ y ) and V = 1 J ( u ∂ η ∂ x + v ∂ η ∂ y ) .

As the incompressibility hypothesis was assumed, the continuity equation is such that:

∂ U ∂ ξ + ∂ V ∂ η ︸ pressure term = 0 (3)

therefore, the governing equations of our computational model, in curvilinear coordinates, are given by Equations (1)-(3).

From the velocity field, obtained by the Equations (1)-(3) and the pollutant transport model provided by:

∂ ∂ τ ( C J ) ︸ temporal term + ∂ ∂ ξ ( U C ) + ∂ ∂ η ( V C ) ︸ convective term = + D [ ∂ ∂ ξ J ( α ∂ C ∂ ξ − β ∂ C ∂ η ) + ∂ ∂ η J ( γ ∂ C ∂ η − β ∂ C ∂ ξ ) ] ︸ diffusive term − K C J ︸ reactive term (4)

where the trajectory of the pollutant concentration in the generated meshes is determined, using the curvilinear coordinate system, described in all blocks.

In the Equation (4), C = C ( ξ , η , τ ) is the pollutant concentration in time t, where t = τ because we are not admitting mesh movement [26], D is the diffusion coefficient and K is the decay coefficient [21] [30].

The numerical code used in the present work refers to the same code used in Cirilo et al. [26] and Saita et al. [30]. Numerical tests on problems involving a single block [26], were presented in the following cases: of parallel plates, of a cavity, and of the arteriosclerosis.

At [30], the mesh used was of the multiblock type, and the velocity field was calculated numerically in accordance with data taken from the literature [31] [32].

Emphasizing that the generalization of the computational code, with its convergence guarantee requirements, from a single block to multiblock is natural. Once the converged velocity is obtained for each block in agreement with adjacent blocks, for reasons of numerical analysis, the multiblock solution is assumed.

Still, the numerical results presented in [26] [30] [33] showed agreement with the literature for several Reynolds numbers [33] [34] [35] [36] [37] by the vortex center location. Finally, the article of [30] validates the code that is used in the present work.

In particular, the parallel plates problem is presented to evaluate the code. Consider a rectangular geometry with eight meters length and a meter height [26] [34] where the flow is predominantly laminar and practically without vortex formation.

The fluid with R_e = 100 is injected into the mesh, with velocity u = 1.0 m∙s⁻¹ and initial conditions for velocity and pressure are considered null. Boundary conditions CIPR, CECO and CNEI are applied to the left, right, top and bottom sides of the geometry, respectively. For the analysis, was considered the mesh refinements [26] as described at Table 1.

The simulations were performed until reaching a steady state, with a tolerance error equivalent to 10⁻³. In solving the linear system, the Gauss-Seidel method was used. For there to be convergence, it was considered Δ t = 10 − 2 for the meshes P₁ - P₃ and Δ t = 5 × 10 − 3 for P₄ and P₅. Considering the mesh P₅, was observed that the speed reaches values close to 1.5 m∙s⁻¹ at the beginning of the simulations [26], but with the progress of the process, this value remains only in the central region of the geometry, as expected. Cirilo et al. [26] present a convergence analysis, in the center of the domain, where for y= 0.5 the maximum analytical velocity is 1.5 m∙s⁻¹ [34]. For mesh refinements, Table 1, the results are presented in Table 2, where Error y = 0.5 = | 1.5 − u y = 0.5 | , u y = 0.5 represents the values obtained numerically in y= 0.5, in each mesh.

To evaluate the two-dimensional meshes obtained in the curvilinear coordinate system, besides the velocity field when solving the Navier-Stokes equations and the dispersion of the pollutant concentration by solving the transport equation, two tests were performed. Observe that the model considers that the problem is infinitely extended in the third dimension.

In the first test, three cases have been proposed, with meshes containing a chimney followed by an obstacle, using different chimneys heights and the obstacle height was fixed. The test aims to verify the vortices appearance, in the blocks, in order to obtain agreement with as proved in the literature [14]. The second test describes a geometry that represents a valley, which portrays a test of environmental interest. Particularly, in this test, the movement and dispersion of pollutant emissions into the atmosphere of an industry located in the Peruvian city of La Oroya were verified.

In this work, it is proposed the use of a curvilinear coordinate system and the multiblock technique, generating meshes that coincide with the physical domain, with good quality elements and with no increase of computational cost. The consistency of the fluid dispersion simulation in the meshes divided into blocks was guaranteed by treating the limits that indicate adjacency between the blocks.

In this way, numerical simulations of the movement and dispersion of the pollutant emissions into the atmosphere were presented in the generated domains. For the discretization of the Navier-Stokes and transport equations were considered the curvilinear coordinates and the finite difference method.

The suggested method was verified in two tests. In the first test, three cases were proposed, with geometries containing a chimney followed by an obstacle, using different chimney heights and the obstacle height was fixed. Qualitatively, the results obtained showed an expected behavior, resulting in the formation of vortexes in the predicted locations of the velocity field, and higher concentration of pollutants accompanying the velocity vectors. Also, in this test, were maintained the same boundary conditions, which allowed us to conclude that the trajectory of the pollutants is affected by the height of the source in relation to the obstacle.

Through the performed tests, were verified that the height of the chimney can be considered a determining factor to describe the dispersion of pollutants, as well as their concentration in the proximity of industrial areas.

The authors are grateful to the Department Mathematics of the State University of Londrina and the Araucária Foundation under grant n˚ 39225, for the encouragement and providing facilities to carry out the present study.

The author declares no conflicts of interest regarding the publication of this paper.

Romeiro, N.M.L., Cirilo, E.R., Natti, P.L., Okamoto, L.M.D. and Julianotti, T. (2021) Chimney Height, a Determining Factor in the Dispersion of Pollutants and Their Concentration. World Journal of Engineering and Technology, 9, 173-193. https://doi.org/10.4236/wjet.2021.91013