2.1. Simplification of the Equation

We will write delta Na by delta y, delta Na by delta z, terms also in this case so that will become sothereifIamonlydoingforonex,Igetthisequation.IfIdotheothers,Iwillgetthisextended equation, we have we have dispersion in all the three directions. This is where we stopped last class.

In this section, we start by introducing the mathematical expression for dispersion in three dimensions: x, y, and z. We denote the change in concentration (Na) with respect to the spatial coordinates y and z. By focusing on just one direction initially (which we denote as x), we can simplify the problem, leading to an equation that describes the concentration dispersion. Later, by including the other dimensions, we extend this equation to account for all directions of dispersion.

Imagine throwing a stone into a lake. The ripples spread out in all directions. Now, if we only look at how the ripples move in one direction, we simplify the situation, even though in reality, the ripples are affecting the entire surface of the water. The same concept applies when simplifying the equations of dispersion.

Now here we make two assumptions, one is a steady state assumption you don’t have to do this but for that Gaussian dispersion model that we generally present we use the steady state assumptions where we say &=0, which means that at any point in time the concentration is the same, at any location you measure it concentration will not change with time. It will be different with space but it will not change with time...

The steady-state assumption simplifies our calculations significantly. By claiming that the concentration does not change over time at any specific location, we can treat the problem as static instead of dynamic. This implies that the conditions (such as pollutant release rate) remain constant during the observation period. For the model to hold true, we must ensure that no other factors affect these attributes over time; otherwise, the assumption would fail.

Think about a tap that pours water into a bucket. If the tap's flow is constant and you look at the bucket at different times, the water level will rise steadily without fluctuating, assuming no water is spilling out. This is similar to our steady-state assumption; as long as conditions remain unchanged, we can predict the water level (concentration) reliably.

Thesecondthingthatwedoissincewealreadyhaveuxbulkflowinginthexdirection.Weneglect thisdxterm,sothisentirethingreducestothissimplerequationthisiswherewestoppedlastclass.

In our analysis, we recognize that there is a bulk flow of air in the x direction, denoted as ux. This means that the transport of the pollutant is dominated by this flow, allowing us to ignore some of the terms in our equations for simplification purposes. By neglecting the dx term, we can focus on the essential aspects of concentration dispersion without the complexities introduced by the flow's variability.

Imagine a crowded room where everyone is walking towards a door. If we consider the flow of people (the ux flow), it may be unnecessary to account for the individual movements in the opposite direction since the majority move towards the exit. Thus, we simplify our focus to the overall movement rather than the minor deviations.

Now, the general solution for this, this is an equation there are solutions are already there. This form, you do separation of variables and do all that...

The next step involves deriving a general solution to the dispersion equation using methods like separation of variables. This approach indicates a structured way to break down the equation into simpler components that can be solved individually, allowing us to combine the solutions later. Constants arise from considering various dimensions, leading to a simplified equation that incorporates all boundary conditions relevant to the problem.

Imagine solving a puzzle. Each piece can represent a different parameter in our equation. By separating and organizing these pieces according to their image, we can effectively complete the entire picture (or equation) without losing sight of how each part interacts.

Now, here, we use an ideal assumption that if you look at this equation, this is very similar to another equation which is called as a Gaussian distribution or a normal distribution...

We draw a connection between the derived equation of concentration dispersion and the well-known Gaussian distribution. This relationship indicates that the dispersion of pollutants can be modeled similarly to the distribution of random variables around a mean value. The characteristic bell-shaped curve of the Gaussian distribution represents variations in concentration with a peak at the central value and tails extending indefinitely. This similarity aids in predicting how pollutants spread in the environment.

Consider how people's heights in a population typically follow a bell curve. Most people are average height, with fewer very short or very tall individuals. This is analogous to how pollutant concentration spreads, with the highest concentration at the center and diminishing concentration further away.

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