5.2 Case Study II: A Two-Dimensional Photochemical Air Quality Model

Uncertainty Analysis of the Reactive Plume Model (RPM-IV)

The Reactive Plume Model, version IV (RPM-IV), is a standard regulatory model used for calculating pollutant concentrations and establishing causal relationships between ambient pollutant concentrations and 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 is often applied for regulatory purposes to calculate the ozone (O$_3$) concentrations at locations downwind of industrial point sources, since high ozone concentrations in the ambient environment lead to adverse health effects. Ozone is primarily formed in the atmosphere through a series of complex chemical reactions involving oxides of nitrogen (NO$_x$) and volatile organic compounds (VOCs) in the presence of sunlight. Some of the major point sources of NO$_x$ and VOCs are industrial units, such as the power plants (NO$_x$ sources) and refineries (VOC sources). The application of RPM-IV helps in establishing a quantitative causal relationship between emissions and ambient pollutant concentrations, which is useful in assessing various control strategies for emission reductions.

However, there are significant uncertainties in developing estimates of the emissions from industrial sources. These uncertainties occur with respect to the amounts of emissions (e.g., total amounts of VOCs and NO$_x$), and with respect to their chemical compositions (or speciations, i.e., the fractions of various chemicals within these groups of compounds). These uncertainties arise due to a variety of reasons: for example, emission estimates are typically derived from hourly averages projected from annual or seasonal averages [38]. Since there could be a significant variation in the load and operating conditions of an industrial unit, emission estimates and chemical compositions for specific days under consideration may differ significantly from the averages, and thus result in significant uncertainties. Hence, an uncertainty analysis that takes into account the emission estimate uncertainties is useful for a better understanding of the effects of control strategies for emission reductions.

5.2.1 Description of RPM-IV

A brief description of the RPM-IV model is presented here; additional details are presented in Chapter 6, which deals with the model uncertainty associated with RPM-IV, and which presents an improved, three-dimensional version of the model.

RPM-IV simulates mechanistically the complex nonlinear photochemistry and dispersion processes occurring in an expanding plume. The nonlinear atmospheric gas phase photochemistry is described by the Carbon Bond IV (CB-IV) mechanism [216], which consists of a set of 95 chemical reactions among 35 surrogate chemical species corresponding to organic bonds and functional groups. Details on the CB-IV mechanism are presented in Appendix C. This model follows the trajectory of an expanding, moving plume and simulates its evolution. In this model, the pollutant mass is initially divided into cells containing equal amounts of pollutants. As the plume expands, the individual cells expand in volume and pollutant mass is transferred across cell boundaries 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 pollutants in each cell undergo chemical transformation governed by the Carbon Bond-IV mechanism.

The equation describing the concentration changes within a cell is given by:

$$% \frac{d c_j^i}{d t} = \left(\frac{d c_j^i}{d t}\right)_{\mb... ...left(\frac{1}{h_j}\cdot\frac{d h_j}{d s}\right)uc_j^i + F_j^i$$ (5.5)

where $c_j^i$ denotes the concentration of chemical species $i$ in cell $j$. $F_j^i$ 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. Expressions for entrainment and detrainment are given by the following equations:

$$% E_j^i = \frac{1}{y_j - y_{j-1}} \left\{\left(\frac{d y_j}{d... ...c^i_{j+1} - \left(\frac{d y_{j-1}}{d t}\right)c^i_{j}\right\}$$ (5.6)

$$% D_j^i = \frac{\partial }{\partial y}\left(K\frac{\partial c... ...1}\left(\frac{{c_j^i}-c^i_{j-1}}{y_j - y_{j-2}}\right)\right\}$$ (5.7)

where $y_{j}$ is the distance from the plume centerline to the far most side of the cell $j$, in the horizontal direction. 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 following equation for expansion of the cells, as the plume travels downwind:

$$E_j^I - D_j^I = 0$$

RPM-IV was selected for evaluation of the SRSM and SRSM-ADIFOR because it is sufficiently complex to represent a wide range of environmental models, and at the same time it is computationally feasible to perform a large number of Monte Carlo simulations with this model. Further, the complex nonlinear photochemistry of this model is employed by a number of photochemical models that are computationally very demanding. Thus, evaluation of the SRSM and SRSM-ADIFOR with RPM-IV could potentially serve as a preliminary test of applicability of this method to a wide range of complex photochemical models.

5.2.2 Uncertainty Analysis of RPM-IV

The present work studies the effect of uncertainty in emission estimates on predicted downwind secondary pollutant concentrations; here, the specific focus is on the uncertainties in the calculated ozone concentrations resulting from the uncertainties in amounts and the chemical composition of the emissions. This study uses emission estimates and field data measured near Marathon oil refinery at Robinson, Illinois, during June and July, 1977 by Sexton et al. [183]. Since there was no characterization of uncertainty, and since the focus of this work is to evaluate the applicability of SRSM-ADIFOR, only representative probability distributions to describe the uncertainties in emissions were assumed.

The following distributions for the emissions of VOCs and NO$_x$ are used: (a) the amounts of VOCs and NO$_x$ released are assumed to have normal distributions with a standard deviation of 20% of the mean value, and (b) the chemical compositions of VOCs and NO$_x$ are assumed to follow a Dirichlet distribution [201]. The Dirichlet distribution satisfies the condition that the sum of mole fractions is unity. According to this distribution the mole fraction of the $i$th compound, $y_i$, is given by:

$$y_i = \frac{x_i}{\displaystyle\sum_{i=1}^{n} x_i}\ ,$$ (5.8)

where $x_i$ is an independent random variable, representing the the amount (in moles) of the $i$th compound; here a normal distribution is assumed for all $x_i$s, with a nominal standard deviation of 40% of mean value. In this case study, the VOC group consists of five subgroups (paraffins, ethylene, olefins, toluene, and xylene), and the NO$_x$ group consists of NO and NO$_2$. Thus, the total number of uncertain parameters for the model is nine (including two parameters representing the total amounts of VOCs and NO$_x$). Thus, a total of 9 uncertain parameters with 7 degrees of freedom were used in this case study; these were represented by 9 srvs.

The output metrics considered are the average ozone concentration in the plume for selected distances downwind from the source.

The original RPM-IV model was implemented in Fortran and obtained from the EPA Exposure Models Library [202]. The derivative code was obtained using the ADIFOR system on RPM-IV code. For Monte Carlo, LHS, and the SRSM, the model was run at selected sample points, whereas for the SRSM-ADIFOR, the derivative model was run.

  

Figure 5.12: Evaluation of ECM: uncertainty in the predicted ozone concentrations at (a) 2 km and (b) 20 km downwind from the source

  
5.2.3 Results and Discussion

From equations  3.10 and 3.10 for the number of unknown coefficients for a second and third order polynomial chaos expansions, for $n$=9, the number of coefficients to be determined is 78 and 364 for second and third order approximations, respectively. For a regression based method, 300 model simulations were selected for a second order approximation, and 600 model simulations were selected for a third order approximation, in order to facilitate regression with an adequate number of model outputs. The estimates of pdfs for these cases were evaluated with the results obtained from 10,000 Monte Carlo simulations and 10,000 Latin Hypercube sampling methods. For the case of the SRSM-ADIFOR, 80 simulations were used for a third order approximation.

Two representative output metrics were considered here: (a) ozone concentration at a downwind distance of 2 km, representative of near-source transport and transformation, and (b) ozone concentration at a downwind distance of 20 km, representative of larger scale transport and transformation. The output pdfs were obtained first using second and third order approximations of the SRSM/ECM. Figure 5.12 shows the pdfs of ozone concentration at a downwind distance of 2 km and of 20 km, as estimated by the ECM, and the conventional Monte Carlo method.

Figure 5.13 shows the pdfs of the predicted ozone concentrations at downwind distances of 2 km and 20 km, as estimated by the regression based SRSM, traditional Monte Carlo and Latin Hypercube Sampling.

  

Figure 5.13: Evaluation of SRSM (regression based): uncertainty in the predicted ozone concentrations at (a) 2 km and (b) 20 km downwind from the source

  

Figure 5.14: Evaluation of SRSM-ADIFOR: uncertainty in the predicted ozone concentrations at (a) 2 km and (b) 20 km downwind from the source

As shown in Figures 5.12 and 5.13, although the regression based-method required significantly fewer runs than the Monte Carlo method, the results agree very closely with the Monte Carlo results. On the other hand, the the predictions of ECM become inaccurate as the order of approximation increases, indicating the lack of robustness in the collocation method. This behavior is more prominent for large downwind distance, indicating that the collocation method may not converge when used with highly nonlinear models (the near-source behavior is expected to be less nonlinear than far-source behavior). On the other hand, a regression based method resulted in similar estimates for both second and third order approximations and was consistent for all ranges of downwind distances. The results indicate that, although the regression methods require a higher number of model simulations for uncertainty propagation, compared to the collocation methods, their robustness makes them a more viable tool for uncertainty analysis of complex environmental models.

For the evaluation of the SRSM-ADIFOR method with RPM-IV, Figure 5.14 shows the uncertainty estimates obtained. As shown in the figure, SRSM/ADIFOR gave closer estimates with only 80 model runs, while 10000 Latin Hypercube samples were not sufficient to achieve agreement with the results from the Monte Carlo methods.