6.3 Formulation of The Reactive Plume Model (RPM)

The Reactive Plume Model, version IV (RPM-IV), is a standard regulatory model used for estimating pollutant concentrations in the atmosphere, resulting from the emissions from point sources such as industrial stacks [153,195]. It uses either point source emission estimates or initial plume concentrations as inputs, and calculates downwind concentrations, as the plume expands. RPM-IV makes use of a diffusion/reaction equation similar in form to the atmospheric diffusion equation (Equation 6.1), but with dispersion parameters that evolve with time. This model considers a control volume that moves downwind along the plume trajectory, and solves the diffusion/reaction equation by considering the nonlinear photochemistry and diffusion of chemical species from one cell into another cell.

  

Figure 6.1: Schematic depiction of the evolution of an atmospheric plume (adapted from Stewart et al.,1981)

In this model, the plume is modeled by a set of ``cells'' consisting of equal masses of pollutants, by assuming a Gaussian distribution of the pollutant mass along the plume centerline. As the plume expands, the individual cells expand in volume correspondingly. The transfer of pollutant mass across cell boundaries is modeled in two phases: (a) an ``entrainment'' phase, where the expanding cell boundaries entrain the pollutants from other cells, and (b) a ``detrainment'' phase, where the pollutants diffuse across cell boundaries due to concentration gradients. Further, the cells are modeled to expand in a manner such that the amount of an inert species remains constant within each cell. The expansion of the boundaries, and the equations governing equations for pollutant concentrations within each cell are as described in the following section.

  

Figure 6.2: Schematic depiction of entrainment and detrainment steps simulated in the RPM (adapted from Stewart et al., 1981)

6.3.1 Initial Physical Dimensions of the Plume

The total width $W(s)$ and the total depth $H(s)$ of the photochemical plume can be estimated as functions of the horizontal and vertical dispersion parameters, as follows:

$$W(s) = f_x\,\sigma_x (s)\ \ \mbox{and} \ \ H(s) = f_y\,\sigma_z (s),$$ (6.2)

where $s$ is the total distance traveled by the plume up to time $t$under consideration, and $f_x$, and $f_z$ are selected as cutoff values of the plume boundaries. Based on the Gaussian distribution of initial plume mass [181], $f_x$ and $f_z$ indicate the amount of pollutant mass that is included in the model boundaries, as functions of standard normal quantiles. A typical value of $f_x = f_y = 4.0$ implies that 95.4% of the total pollutant mass is included within the plume boundaries.

For a plume described by $M$ cells in the horizontal direction, with only one cell in the vertical, at the start of the simulation each cell contains equal amount of pollutant mass. Since the pollutant mass is assumed to follow a Gaussian distribution, the width of the cell $j$, $w_j$, can be calculated using the recursive equation

$$\frac{2}{M} = \mbox{erf}\left(\frac{w_j}{\sqrt{2}\sigma_y}\right) - \mbox{erf}\left(\frac{w_{j-1}}{\sqrt{2}\sigma_y}\right),$$ (6.3)

where $w_j$ is the width of the $j$th cell [195].

6.3.2 RPM-IV Model Equations

Based on mass balance, the governing equation of the Reactive Plume Model is given by:

$$\frac{d c_j^i}{d t} = \underset{A}{\underbrace{ \underset{\ ... ...rac{d h_j}{d s}\right)uc_j^i}}} +\overset{D}{\overbrace{F_j^i}}$$ (6.4)

Here, the term $(A)$ describes the change in pollutant concentrations due to chemical transformation, term $(B)$ the dilution as the $j$th cell expands in the horizontal, term $(C)$ the dilution as the $j$th cell expands in the vertical, and term $(D)$ the net flux of the $i$th contaminant into the cell.

In this equation, $c_j^i$ denotes the concentration of chemical species $i$in cell $j$, $u$ denotes the speed at which the cells travel downwind, and $w_j$ and $h_j$ denote the width and height of the $j$th, respectively. Additionally, $F_j^i$ is the net flux into the cell, and is given by $F_j^i = E_j^i - D_j^i$, where $E_j^i$ is the entrainment as the cell expands, and $D_j^i$ is the detrainment due to the diffusion of the species across the boundaries of the cell. Figure 6.2 describes the process of entrainment and detrainment in detail. Expressions for entrainment and detrainment are given by the following equations:

$$E_j^i = \frac{1}{y_j - y_{j-1}} \left\{\left(\frac{d y_j}{d ... ...c^i_{j+1} - \left(\frac{d y_{j-1}}{d t}\right)c^i_{j}\right\}$$ (6.5)

$$D_j^i = \frac{\partial }{\partial y}\left(K\frac{\partial c}... ...1}\left(\frac{{c_j^i}-c^i_{j-1}}{y_j - y_{j-2}}\right)\right\}$$ (6.6)

where $y_{j}$ is the distance from the plume centerline to the far most side of the cell $j$, in the horizontal. An additional condition is imposed on the expansion of the cells: the cells are allowed to expand in a way such that the amount of an inert species remains constant within each cell, i.e., no net transfer of an inert species occurs from a cell. This condition results in the equation $E_j^I - D_j^I = 0$, for the expansion of the cells, as the plume travels downwind. This condition leads to a recursive relationship to compute $K_j$, as follows:

$$\left(\frac{d y_j}{d t}\right)C^I_{j+1} - \left(\frac{d y_{j-... ...1}\left(\frac{{C_j^I}-C^I_{j-1}}{y_j - y_{j-2}}\right)\right\}$$ (6.7)

where $$C^I_j$$ is the concentration of an inert species $I$ in the $j$th cell. Further, the condition of zero concentration gradient at the plume centerline results in the equation:

$$K_1 = \left(\frac{C^I_2}{2} \displaystyle\frac{\displaystyle y_2 - y_0}{C^I_2 - C^I_1}\right)\frac{d y_1}{d t}$$ (6.8)

The values of $K_j$ can then be recursively obtained using the above two equations. Once $K_j$s are computed, Equations 6.4, 6.5, and 6.6 can be used to compute the concentration in each cell at different times.

In RPM-IV, the nonlinear plume photochemistry is modeled through the carbon-bond mechanism, version IV (CB-IV) [216]. CB-IV describes the complex non-linear gas-phase atmospheric photochemistry through a set of 95 reactions among 35 surrogate chemical species corresponding to organic bonds/functional groups [216]. This mechanism lumps similarly bonded carbon atoms, resulting in a condensed mechanism of that is used widely in the regulatory photochemical models. A brief description of the CB-IV mechanism, and the chemical reactions considered, is presented in Appendix C.

6.3.3 Limitations of the RPM-IV

Figure 6.1 presents a schematic depiction of the physical structure of a plume as it evolves from a stack, and travels downwind. The assumption of uniform vertical concentration in the RPM (hence, only one ``well mixed'' cell in the vertical) is valid only at distances much greater than $X_{TD}$ from the point source. This poses a limitation, since the typical touchdown distance could vary between a few hundred meters to a few kilometers, depending on the meteorological conditions, the height of the stack, and the exit velocity at the plume source. When the value of $X_{TD}$is large, the well-mixedness assumption is not valid at the vicinity of the plume, thus lessening the main advantage of using a plume model to study the ``sub-grid'' scale, local phenomena. In such cases, if the RPM is used with the uniform vertical mixing assumption, significant ``errors'' could result in the model calculations.

In order to address this limitation of the RPM-IV, a corresponding ``three-dimensional'' version of the RPM, called RPM-3D, is developed as part of this work. This RPM-3D simulates the evolution of a plume by dividing the plume cross-section into rectangular regions consisting of equal initial pollutant mass; the plume cross section is divided into columns of cells, as in the case of RPM-IV, and each column is further subdivided into rows.

  

Figure 6.3: Discretization of the plume cross section by RPM-IV (left) and RPM-3D (right)

Figure 6.3 illustrates the physical structure of the plume cross section in the RPM-IV and RPM-3D. In the figure, R1 is the region containing the cell closest to the plume centerline, R2 contains the ground-level cell along the plume centerline in the horizontal, and R3 contains the centerline cells with respect to both the horizontal and the vertical. Here, R1 corresponds to the representation used by RPM-IV, whereas R2 and R3 correspond to representations used by RPM-3D. In typical cases, the extreme pollutant concentration occurs along the plume centerline, and the pollutant concentration decreases as the distance of the cells from the centerline increases. Hence, the extreme pollutant concentration (e.g., concentration of ozone) calculated by the RPM-IV is an average over the region R1. This provides a concentration for the estimation of exposure to ozone. However, as shown in the figure, the region R2 corresponds to the most likely ground level maximum ozone concentration (typically covering the region from the ground to up to less than a hundred meters), whereas the region R3 corresponds to the maximum ozone level concentration at the plume centerline. Clearly, the RPM-3D provides estimates of pollutant concentrations at a detailed spatial resolution that is appropriate for estimating human exposure.