1.1. Transport of Pollutants - Gaussian Dispersion Model

So, our goal is to model the system to predict rho A1 as a function of x, y, z and time, this is our general prediction. If you are trying to invoke the mass balance and trying to develop mathematical models. So, here two things may be happening, its rate in and rate out. So rate of accumulation equals rate in minus rate out plus there is no reaction (we are assuming no reactions here).

In this part, we establish that our goal is to model the transport of pollutants, specifically how the concentration of a pollutant (denoted as rho A1) changes over time and space (x, y, z). The basic principle behind this is based on mass balance, which involves calculating the rate at which the pollutant enters and exits a given volume. The equation states that the change in pollutant concentration within that volume is equal to the amount coming in minus the amount going out, presuming no reactions occur (like degradation or chemical interactions). This simplifies our model, as we only focus on physical dispersion instead of chemical processes.

Imagine a bathtub filling with water. The rate of water entering the bathtub can be thought of as the 'rate in,' while the rate of water draining out represents the 'rate out.' If the water is not spilling over the edge (no chemical reactions happening), the amount of water (the concentration of pollutants, in our case) filling up the tub can be predicted easily by measuring how fast it's filling and draining.

So dispersion models can be of two different kinds. One is called an Eulerian model, which is a fixed reference frame. What this means is, if I am modeling this room here. I am watching from here. Lagrangian model on the other hand is that you are moving with the fluid that is the frame of reference is that body of fluid.

This section introduces two major types of dispersion models: Eulerian and Lagrangian. The Eulerian model observes changes from a fixed point in space, like watching the weather from a stationary station. In contrast, the Lagrangian model follows specific fluid particles as they move through space and time – akin to riding in a river and observing your surroundings as you float along. Each model has distinct applications and helps in visualizing and quantifying how pollutants disperse differently depending on the frame of reference.

Picture watching a parade from the sidelines versus being part of the parade. In the Eulerian perspective (sidelines), you see the overall event unfold, while in the Lagrangian perspective (participating), you only see what's around you as you move along with the flow of the parade. Both perspectives provide valuable information, but focus on different elements of the same event.

So here dispersion model is set up as the most commonly used model. This thing is what is called the Lagrangian model. We are now looking at the plume (we are looking at the entire system), but we are also seeing that when we are talking about the z and y and all that and the dispersion it is the reference to this particular, it is not with the reference to a fixed reference frame.

The focus here is on how we manage and observe pollutant dispersion through the Lagrangian model by examining the behavior of a plume. The plume is essentially a cloud of pollutants released into the environment, and we’re more interested in how it spreads over time and in three dimensions rather than how a fixed location experiences its impact. This highlights the dynamic nature of dispersion, emphasizing the need to model both vertical and horizontal spreading in relation to the plume's movement.

Think of a smoke signal rising from a campfire. Watching the smoke as it rises and spreads is similar to how the Lagrangian model views the plume of pollutants. If you’re walking with the smoke as it travels, you can observe how it curls and disperses over the landscape, giving you a better understanding of how air pollution behaviors change with environmental conditions.

So when we write the general equation, we have written down dA1. Here I will change it to zero. I will derive this equation for you in a minute. Let us say that we have a small volume. We take a three-dimensional volume. This is delta x, this is delta y, this is delta z.

Here, we start delving into the mathematical formulation of the dispersion model. We use a three-dimensional volume represented by changes in the x, y, and z directions, denoted as delta x, delta y, and delta z. This allows us to systematically analyze how pollutants change within a prescribed volume, accounting for their movement and spreading due to turbulence and flow dynamics. It leads to the development of differential equations that describe the rate of change of pollutant concentration over time.

Imagine measuring the extent of a puddle spreading outwards on a flat surface. You would look at small sections of the puddle ('delta x', 'delta y', 'delta z') to determine how quickly the water is moving into new areas. This local measurement can be related back to understand the whole layout of the puddle at larger scale, just as this approach helps us model the dispersion of pollutants in the environment.

When will concentration change? Let us say if I have a plume here, so I am measuring concentration at this point. I would like to find out what is the concentration at this location which has a certain particular z, particular x, and some y. And at this point, if I want to measure concentration it will only be an unsteady state.

In this part, we address the conditions under which we anticipate a change in the concentration of pollutants over time (unsteady state). If the concentration at a certain point remains constant over time, we refer to that as a steady state. However, if conditions like wind speed, source emissions of the pollutant, or environmental conditions vary, we will observe fluctuations in concentration. Understanding these conditions is crucial for predicting pollutant behavior accurately.

Imagine standing by a riverbank and watching the water flow. If the river’s flow stays constant (same amount of water entering and exiting), changes at any point will be minimal, reflecting a steady state. However, if heavy rain suddenly causes more water to rush in, it alters the river’s flow downstream, illustrating how external changes can lead to an unsteady state and fluctuations in water levels.

--