4.1. Formulation of Normal Distribution

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.

The steady state assumption means that the concentration of a substance at any location does not vary over time. This is important in the context of Gaussian dispersion models, which are often used to model how pollutants spread in the air. If the concentration remains constant over time, we can better predict how the pollutant will behave in different spaces. For this to hold true, other factors such as emission rates need to be stable.

Imagine standing in a room and spraying air freshener. If the air freshener is continuously released and you stand there for a while without moving, the smell concentration around you will remain fairly constant, which reflects the steady state assumption.

Nothing should change with time. If something if any of the parameters in this model changes with time, then this is not true. You cannot use a steady-state assumption. So, which means it is an assumption that in an environment nothing is constant everything is changing.

When making the steady state assumption, it is crucial that emission rates and other factors impacting concentration remain constant. If, for example, the source of pollution suddenly increases or decreases its output, the assumptions of the model will no longer hold true, leading to inaccurate predictions about pollutant dispersion.

Consider a factory that constantly releases smoke into the air. If the factory decides to double its output, the concentration of smoke downwind will change, and the previous predictions made under a steady state assumption would no longer be valid.

So we are doing all of that in one shot what we are saying is there using a mass conservation in the plume if you take the entire volume of the plume, the entire volume is coming from the source so we are saying Q is the rate of pollutant release equals u multiplied by dy and dx.

In modeling pollutant dispersion, it's essential to consider all boundary conditions that relate to mass conservation. The pollutant mass flow rate (Q) is linked to its velocity (u) and the areas of dispersion (dy and dx). This relationship allows us to understand how much pollutant is released over a certain area in space and helps in defining the overall behavior of the plume through equations.

Think of a garden hose spraying water. The rate at which water flows out of the hose (Q) will help you determine how much area gets wet (dy and dx) and how fast the water spreads. If the flow rate doubles, obviously, a larger area gets wet faster.

Now to make it look like this you have to do some transformation so one of the transformation is this one this one we do sigma_y and sigma_z into 2 Dyx divided by ux and so on.

The Gaussian distribution describes how pollutants disperse in the air over time and space. To relate our dispersion model to this distribution, we need to transform certain parameters like sigma (which describes spread in y and z directions) and relate them to diffusion coefficients and wind velocity. This transformation helps fit our model into a familiar mathematical framework that accurately represents how concentration varies in different directions.

Imagine baking a cake where you need to adjust the recipe quantities to fit a smaller or larger pan. Similarly, we adjust parameters in our model to ensure it accurately reflects the dispersion characteristics of pollutants.

If I am plotting instead of this if I plot instead of, f of y what I am plotting this rho A1 as the function of z or either.

When observing how pollutants disperse, it's important to visualize their concentration differences at various points. By mapping concentration (rho A1) as a function of y or z, we can see where the highest concentrations occur in relation to the plume's shape and dispersion characteristics. This helps us understand the distribution patterns of pollutants in the environment.

Think of a foggy day. The density of fog varies across the street. In one part, it may be very thick (high concentration), while in another, it may hardly be visible (low concentration). By mapping this out, we can better understand where to be cautious while driving.

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