$\frac{\partial }{\partial t}\left(\rho \varphi \right)+\text{div}\left(\rho u\varphi \right)=\text{div}\left({\Gamma }_l\text{grad}\varphi \right)+S_{\varphi }$ (1)

The four terms in the general differential Equation (1) are the unsteady term, the convection term, the diffusion term and the source term respectively. The dependent variable $\varphi$ can stand for a variety of different quantities, such as the mass fraction of a chemical species, the enthalpy or the temperature, a velocity component, the turbulence kinetic energy, or a turbulence length scale.

Accordingly, for each of these variables, an approximate meaning will have to be given to the diffusion coefficient Γ_. and the source term S.

Note:

Not all diffusion fluxes are governed by the gradient of the relevant variable. The use of $\text{div}\left(\Gamma \text{grad}\varphi \right)$ as diffusion term does not, however, limit the general $\varphi$ equation to gradient driven diffusion processes. In fact, Γ can even be set to zero.

The flow field should satisfy the mass conservation or the continuity equation;

$\frac{\partial \rho }{\partial t}+\text{div}\left(\rho u\right)=0$ (2)

The Cartesian-tensor form of Equation (1) and Equation (2) are:

$\frac{\partial }{\partial t}\left(\rho \varphi \right)+\frac{\partial }{\partial x_j}\left(\rho u_j\varphi \right)=\frac{\partial }{\partial x_j}\left({\Gamma }_{.}\frac{\partial \varphi }{\partial x_j}\right)+S_{.}$ (2a)

$\frac{\partial \rho }{\partial t}+\frac{\partial }{\partial x_j}\left(\rho u_j\right)=0$ (2b)

$j$ can take the values 1, 2, 3. i.e.;

$\frac{\partial }{\partial x_j}\left({\Gamma }_{.}\frac{\partial \varphi }{\partial x_j}\right)=\frac{\partial }{\partial x_1}\left({\Gamma }_{.}\frac{\partial \varphi }{\partial x_1}\right)+\frac{\partial }{\partial x_2}\left({\Gamma }_{.}\frac{\partial \varphi }{\partial x_2}\right)+\frac{\partial }{\partial x_3}\left({\Gamma }_{.}\frac{\partial \varphi }{\partial x_3}\right)$

$\frac{\partial }{\partial x_1}\left({\Gamma }_{.}\frac{\partial \varphi }{\partial x_1}\right)+\frac{\partial }{\partial x_2}\left({\Gamma }_{.}\frac{\partial \varphi }{\partial x_2}\right)+\frac{\partial }{\partial x_3}\left({\Gamma }_{.}\frac{\partial \varphi }{\partial x_3}\right)=0$; is the Helmhorltz equation for scalar fields

$\frac{\partial }{\partial x_j}\left(\rho u_j\right)=\frac{\partial }{\partial x_1}\left(\rho u_1\right)+\frac{\partial }{\partial x_2}\left(\rho u_2\right)+\frac{\partial }{\partial x_3}\left(\rho u_3\right)$

The recognition that all the relevant differential equations for heat, mass transfer, fluid flow, turbulence and related phenomena can be thought of as particular cases of the general $\varphi$ equation is an important time-saving step. Since it can be repeatedly used for different meaning of $\varphi$ along with appropriate expressions for Γ and S, and of course with appropriate initial and boundary conditions. The concept of the general $\varphi$ equation enables us to formulate a general Numerical method and to prepare general purpose computer programs.

The velocity components are governed by the momentum equations. This momentum equations are special cases of the general differential equation for $\varphi$ where $\varphi$ and Γ and other parameters are replaced; i.e. $\varphi =u$, $\Gamma =\mu$ and so on.

The difficulty in the calculation of the velocity field lies in the unknown pressure field. The pressure gradient forms part of the source term for a momentum equation. If the pressure field is given, the momentum equation becomes easy to solve.

From the general differential equation for velocity field problems:

$\frac{\partial }{\partial t}\left(\rho \varphi \right)+\frac{\partial }{\partial x_j}\left(\rho u_j\varphi \right)=\frac{\partial }{\partial x_j}\left({\Gamma }_{.}\frac{\partial \varphi }{\partial x_j}\right)+S_{.}$

Since the flow field must satisfy the continuity equation:

$\frac{\partial \rho }{\partial t}+\frac{\partial }{\partial x_j}\left(\rho u_j\right)=0$

The general differential equation becomes:

$\rho \frac{\partial \varphi }{\partial t}+\rho u_j\frac{\partial \varphi }{\partial x_j}=\frac{\partial }{\partial x_j}\left({\Gamma }_{.}\frac{\partial \varphi }{\partial x_j}\right)+S_{.}$ (3)

$\varphi$ can be: a chemical species, diffusion flux, heat flux, viscous stress, etc.

For a given differential equation, the required discretization equation can be derived in many ways. Here, we shall outline a few common methods and then indicate a preference:

The control volume formulation satisfies the integral conservation of quantities such as mass, momentum, and energy over any group of control volumes and of course, over the whole calculation domain.

The difficulties faced by standard F.E.M. were removed by using a F.E.M based on a control volume [30] .

Steps:

1) For the triangular grid the dependent variables are calculated for grid points that lie at the vertices of the triangles. The discretization equations are formed by the control volume method (i.e. the differential equation is integrated over the typical control volume as shown below). The control volumes are constructed by joining the centroid of each triangle to the midpoints of the sides of that triangle ( Figure 1 ). The discretization equation is formed by adding the contribution of these elements to the integral conservation for the control volume.

2) The shape function describing the variation of $\varphi$ over an element is needed to calculate the flux across the control-volume faces that fall within the element. The standard shape function for the triangular element is;

$\Phi =a+bx+cy$ (4)

This gives birth to difficulty; so, the above shape function is unacceptable. The alternative proposed by Baliga and Patankar [30] , is the shape function:

$\Phi =A+B\exp \frac{\rho uX}{\Gamma }+CY$ (5)

u: is the resultant velocity in the element.

X: is the coordinate pointing in the direction of the resultant velocity.

Y is the direction normal to X.

A, B, and C are found in terms of the three values of Φ at the vertices of the triangle.

For low peclet numbers Equation (5) boils down to Equation (4) which is the appropriate shape function for conduction problems.

It is through Equation (5) that the spirit of the exponential scheme has been introduced into the finite element method.

Note: F.E.M. based on Equation (5) produces much less false diffusion than does other formulations.

3) The issue of the staggered grid is handled by calculating the pressure on a grid that is different from the grid used for all the other variables. The pressure is calculated at the vertices of “macrotriangles” which are shown by small circles. Figure 2 , shows each macrotriangle divided into four subtriangles. The subtriangles form the grid for the velocity components and all other variables except pressure.

Note: peclet number is directly proportional to the distance between grid points.

4) A sequential solution algorithm is formulated. The pressure equation and the pressure-correction equations are derived from the continuity equation written for a control volume defined by the macrotriangle.

The control volume based-F.E.M. represents a logical and effective extension of discretization.

Within the scope of an experimental method (Environmental Wind tunnels) [33] [34] , have developed mathematical models describing traffic induced pollution in cities.

$\frac{u\left(z\right)}{u_{ref}}={\left(\frac{z}{z_{ref}}\right)}^{\alpha }$ (6)

It expresses the wind velocity profile, for modest (4 m/s and up) wind speed conditions.

it is the prerequisite for E.W.T modelling [27] .

$\frac{u}{u_{\ast }}=\frac{1}{k}\ln \frac{z-d_0}{z_0}$ (7)

$u_{\ast }=\sqrt{\frac{\tau }{\rho }}$

Note: d₀ and z₀ are determined by the building configurations.

The earliest ones [33] , assumed a relationship between d₀ and z₀ and average roughness height H such as:

$z_0=0.15H$, and $d_0=0.85H$ (8)

Equation (7) was solved, at a height z 10 to 15 times above z₀ but within the logarithm limits and came out with the expression:

$u_{\ast }=\frac{k\cdot u\left(h\right)}{\ln \left[\left(h-d_0\right)/z_0\right]}$ (9)

Two parameters (coefficients) for the classification of roughness in terms of building arrangements were found and defined as:

${\lambda }_{ar}=\Sigma$ of areas covered by buildings/total urban area

${\lambda }_{fa}=\Sigma$ of average building areas normal to the wind/total urban area

$z_0=0.25H\cdot {\lambda }_{ar}$ (10)

$\frac{P_{Tm}}{P_{Wm}}=\frac{P_{Tn}}{P_{Wn}}$ (11)

$m$: model; $n$: full size

$P_T$: Energy caused by moving traffic

$P_W$: Energy caused by wind

$P_T$ is given per unit street length, by the expression:

$P_T\frac{\rho C_{DT}{A}_{T.}{n}_{T.}{u}_T^3}{B\cdot H}$ (12)

where $\rho$ is the density of air, $C_{DT}$ is the drag coefficient of the (average) vehicle, $A_T$ is the frontal area, and $u_T$ its velocity. $n_{T.}$ is the number of vehicles per unit length.

For $P_W$ they obtained:

$P_W=\tau \frac{\Delta u}{\Delta z}\approx \frac{\rho u_{\ast }^2}{H}u\left(H\right)=\frac{\rho \cdot c_{fh}\cdot u_{ref}^3}{H}$ (13)

where the shear velocity $u_{\ast }=\sqrt{c_{fh.}{u}_{ref}}$ has been expressed through the friction coefficient $c_{fh.}$ consequently, eq. (11) requires that the ratio (the modelling law):

$P_T=\frac{C_{DI}{A}_{T.}{n}_{T.}}{c_f\cdot B}\frac{u_T^3}{u_{ref}^3}$ must be the same in model and prototype.

Here the model considers pollution from different sources (traffic, point sources, etc.). If the quantity to be determined is a concentration from a line source of traffic, then the tunnel experiment is conducted with an emission rate $E_m$ per unit length of the line source, which yields the dimensionless concentrations:

$C^{\ast }\left(\alpha ,u_{ref}\right)=\frac{c_m\left(\alpha ,u_{ref}\right)}{E_m}{u}_{mref}\cdot L_m$ (14)

$c_m\left(\alpha ,u_{ref}\right)$: measured concentration;

$u_{mref}$: reference velocity for the model;

$L_m$: characteristic length.

Model emission $E_m$ has the dimension $c_o\cdot q$, where $c_o$ is the concentration in the emitted gas flow rate $q$ per unit length, which is selected to yield concentration $c_m$ in the model, which can be measured conveniently with the available analysis equipment.

A basic study for street canyons is shown in Figure 3 , with the following considerations [34] :

(B₁/H = 1; B₂/H = 1) placed at a distance of SI (SI/H = 1) apart.

The wind direction is perpendicular to the building rows, as shown below.

The relative speed is corrected by $\sqrt[3]{a}$. The important combination $\sqrt[3]{a\cdot \frac{v}{u}}$

postulated as factor in the modelling law above produced correct scaling of wind tunnel data ( $a$ represents a scaling factor for the traffic density). Further, the

result show that the effect of the combination $\sqrt[3]{a\cdot \frac{v}{u}}$ is linear, i.e.:

$c^{**}\left(\alpha ,u_{ref},v\right)=c^{*}\left(x,y,z,\alpha \right)\left[1-\varphi \cdot a^{1/3}\frac{v}{u}\right]$ (15)

where $c^{*}\left(x,y,z,\alpha \right)=\frac{C\left(x,y,z,\alpha \right)}{E}L\cdot u_{ref}$ is the dimensionless concentration for

the case without moving vehicles ( $v=0$) calculated in the usual way by means of eq. (14). Consequently the effect of traffic amounts to a reduction of the velocity by a term $\varphi \cdot a^{1/3}\cdot v$ in calculating the effect of moving traffic.

Ibrahim et al. [32] have presented variations in pollutant concentration within street canyons using a numerical approach. A study of vehicle exhaust dispersion within different street canyons models in an urban ventilated by cross-wind is conducted at this work to investigate how pollution dispersion is affected by wind speed, building height to width ratios, street and building geometries, and canyon street number. For all cases, cars exhaust was represented by a line source emitting CO₂ and pollution dispersion was studied in two and three dimensions, airflow and the street canyons were assumed thermally isolated. The FLUENT Computational Fluid Dynamics (CFD) software package was used with a standard k-ε, RNG k-ε and sstk-ω turbulence models to simulate different street canyon models.

The main parameter to be calculated is the non dimensionless parameter

$K=\left(C{U}_{\infty }h\right)/\left({\stackrel{\dot{}}{Q}}_{c_{02}}/l\right)$ (16)

$C$ is the volume fraction of pollutant, $h$ is the building height, and ${\stackrel{\dot{}}{Q}}_{c_{02}}/l$ is the emission source strength per unit length. For outlet flow, an outflow boundary condition was assumed.

For this study, Figure 4 shows the contour of the simulated concentration as is displayed from Fluent showing the flow pattern of CO₂ in the atmosphere and around the buildings in the case with an obstacle. Here we are presenting the dimensionless concentration and not the molar concentration as above.

Similarly, Figure 5 gives a contour for a wind speed of 0.8 m/s, as simulated in this study. The dimensionless concentration displayed from Fluent showing the flow pattern of CO₂ in the atmosphere and around the buildings in the case with an obstacle. Here we are presenting the dimensionless concentration with a velocity of 0.8 m/s for the gases.

Furthermore, Figure 6 displays the contour for the values for a wind speed of 0.5 m/s, simulated in Fluent.

Another wind speed of interest is 0.1 m/s. Shown in Figure 7 , its contour of the simulated concentration is displayed from Fluent, showing the flow pattern of CO₂ in the atmosphere and around the buildings.

The above simulations were performed for the CAMPUS model (path). The contours shows that the intensity of the dispersion depends on the wind speed.

1) The general form of the governing equations:

$\frac{\partial \rho }{\partial t}+\frac{\partial u_i}{\partial x_i}=0$

$\rho \frac{\partial u_i}{\partial t}+\rho \frac{\partial }{\partial x_j}\left(u_j{u}_i+\overline{{u^{\prime }}_i{u^{\prime }}_j}\right)=-\frac{\partial P}{\partial x_i}+\frac{\partial }{\partial x_i}\left({\tau }_{ij}\right)$

$\frac{\partial C^{\alpha }}{\partial t}+u_j\frac{\partial C^{\alpha }}{\partial x_j}=\frac{\partial }{\partial x_j}\left(\left(D^{\alpha }+\frac{{\nu }_t}{S{c}_t}\right)\frac{\partial C^{\alpha }}{\partial x_j}\right)$

The above equations are respectively; the mass, momentum and species transport of turbulent airflow conservations, which are written in Cartesian tensor notations in general form. $u_i$ and $P$ are the fluid mean velocity and pressure, ${u^{\prime }}_i$ is the fluctuating velocity, $\rho$ represents fluid density. In the species transport Equation, $C^{\alpha }$ is the concentration of pollutant species α and $D^{\alpha }$ is its diffusivity, ${\nu }_t$ is the turbulence eddy viscosity, and $S{c}_t$ is turbulence Schmidt number.

The term $\rho \overline{{u^{\prime }}_i{u^{\prime }}_j}$ is the time-averaged rate of momentum transfer due to turbulence. The Reynolds stress tensor ${\tau }_{ij}=-\overline{{u^{\prime }}_i{u^{\prime }}_j}$ is given by

$\overline{{u^{\prime }}_i{u^{\prime }}_j}=\frac{{\mu }_t}{\rho }\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right)+\frac{2}{3}k{\delta }_{ij}$ (17)

The turbulent kinetic energy k is defined as half the trace of the Reynolds stress tensor such that

$k=\frac{1}{2}\sqrt{\overline{u_i{u}_j}}$

2) The k-ε model: To determine the turbulent velocity and the length scales, the k-ε model (two equation model) is used that solves one transport equation for turbulent kinetic energy k and another for dissipation rate ε and written as:

$\rho \frac{Dk}{Dt}=\frac{\partial }{\partial x_j}\left[\left(\mu +\frac{{\mu }_t}{{\sigma }_k}\right)\frac{\partial k}{\partial x_j}\right]+{\mu }_t{S}^2-\rho \epsilon$ (18)

$\rho \frac{D\epsilon }{Dt}=\frac{\partial }{\partial x_j}\left[\left(\mu +\frac{{\mu }_t}{{\sigma }_{\epsilon }}\right)\frac{\partial \epsilon }{\partial x_j}\right]+\frac{\epsilon }{k}\left(c_{1\epsilon }{\mu }_t{S}^2-\rho c_2\epsilon \right)$ (19)

3) The sstk-ω turbulence models: The sstk-ω model on the other hand, has the capability of addressing some of the deficiencies of the k-ε model by solving the standard k equation but uses specific dissipation rate ω as a length-determining factor and written as

$\rho \frac{Dk}{Dt}={\tau }_{ij}\frac{\partial U_i}{\partial x_j}-\rho {\beta }^{*}{f}_{{\beta }^{*}}k\omega +\frac{\partial }{\partial x_j}\left[\left(\mu +\frac{{\mu }_t}{{\sigma }_k}\right)\frac{\partial k}{\partial x_j}\right]$ (20)

$\rho \frac{D\omega }{Dt}={\alpha }_{\omega }\frac{\omega }{k}{\tau }_{ij}\frac{\partial U_i}{\partial x_j}-\rho {\beta }^{*}{f}_{{\beta }^{*}}k{\omega }^2+\frac{\partial }{\partial x_j}\left[\left(\mu +\frac{{\mu }_t}{{\sigma }_{\epsilon }}\right)\frac{\partial \epsilon }{\partial x_j}\right]$ (21)

${\mu }_t=\rho c_{\mu }\frac{k^2}{\epsilon },\epsilon ={\beta }^{*}\omega k$ (22)

The principal parameters used in the simulations where carefully selected from literature. The choice of the turbulence model was made based on information gathered from the literature, and others are tested and validated in the course of the study. Table 1 summarizes these parameters.

Measurements on the site were obtained, using the gas tester [KIMO-KIGAZ 700 (060322)]. The tester was set to record concentrations after each point following a click. The values were recorded accordingly and printed at the end of each session. The test was carried out during the morning rush hours between 7:00 am to 9:00 am and evening rush hours, 5:00 pm to 7:00 pm, for five consecutive days. Values were recorded at points corresponding to those used for the numerical simulations and are tabulated. Each measurement is taken five times for both morning and evening sessions, making ten records per point and the average calculated. While measurements where going on, vehicles plying around the campus were being recorded.

The amount of CO₂ emissions emitted from automotive vehicles, can be calculated using the following formula [6] . Automobiles were not classified by type, therefore average emission factor for small cars, medium cars and large cars are calculated and used.

$\begin{array}{c}C{O}_{2}emission=vol\left(unit/hour\right)\times streetlength\left(km\right)\\ \times emissionfactor\left(gC{O}_2/km\right)\end{array}$ (23)

Changes in the carbon dioxide emission field were investigated in campus A, ie in the main campus of the University of Douala which on the north is facing the busy street from «Ange Raphael» to the Cocoa processing industry known as «SIG Cacao», on the left side is a less busy street to campus B through st. Thomas Aquinas university parish, and also runs behind the campus, on the right is an earth road leading into the neighboring surroundings. Opposite the campus and its surroundings are private buildings for living and commercial activities. The campus covers about (424 m × 172 m) of which large part is occupied by the faculty buildings, classroom blocks, University restaurant, health center, and sporting play ground. The campus comprises 25 buildings of varying sizes, while the surrounding buildings are many and of varying sizes and heights. The configuration of the campus is shown in Figure 8 . The paths (models) are designated with respect to the main University service found on the path, e.g. CAMPUSRESTAU, is named as a result of the fact that the University restaurant building is on the chosen path. Figure 9 shows a top view of the campus outlook, which is measured and designed in AutoCAD software.

In this study, observations on the series of buildings situated in the main campus of the University of Douala, will enable to estimate the concentration of CO₂ with the aid of the software Fluent and compare with measured values for five selected paths. The CAMPUS model (path) being the baseline, with a street canyon of 42 walls. Measurements were taken at each wall center. This study was done basing on the following major assumptions: