1. Steady State Assumption

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 $\frac{dC}{dt} = 0$, 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 is a fundamental concept in modeling where the concentration of a substance remains constant over time. In this context, $\frac{dC}{dt} = 0$ implies that at any given point in the environment, if we measure the concentration of a pollutant, it stays the same (does not change as time progresses). However, concentration can still vary by location, meaning that while some places may have more of the pollutant, it won’t increase or decrease over time at those specific locations.

Imagine a bathtub with the tap running while the drain is also open. If the water level remains constant (steady state), it means that the water flowing in equals the water flowing out, leading to a consistent level over time, although the location of the water will change until the tap is turned off.

It will be different with space but it will not change with time, ok. So, if you are looking at it in a plume nothing is going to change, which means for this to be true, everything else has to be true: the emission has to be constant; the properties have to be constant.

For the steady state assumption to hold true, certain conditions must be met. The emission of pollutants into the air must be constant, meaning the source of pollution isn't varying with time. Similarly, the physical properties of the pollutants and the environment (like wind speed and temperature profiles) must also remain unchanged. If any of these factors fluctuate over time, the steady state assumption is invalidated, and we can no longer consider concentrations steady.

Think about the air quality in a factory. If the machines are running at a constant rate and emissions are steady, the air quality measurements will remain steady too. However, if one day a machine starts malfunctioning or if workers increase production, emissions will vary, making the air quality different from previous measurements.

If something changes... 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.

The essence of the steady state assumption is that in a real-world scenario, conditions can continuously change. Therefore, to simplify calculations and predictions, researchers often rely on average values of various factors and their standard deviations. This means they accept that fluctuations exist, but they seek to understand the overall trend or typical conditions through averaging.

Consider measuring the temperature in a bakery. While the temperature might fluctuate every minute due to ovens being opened, on average, the temperature during baking remains around a certain value. This average helps bakers know when to expect the best results for their products, despite the fluctuations.

Now here we neglect the dx term, so this entire thing reduces to this simpler equation.

When we apply the steady state assumption, we can simplify complex equations by neglecting certain terms that become irrelevant in a steady state scenario. By reducing our equations, we can focus on the most significant factors that influence pollutant dispersal, making it easier to understand and model how pollutants spread in different environments.

Think of simplifying a recipe by ignoring optional spices that won't change the dish's overall flavor. You can focus on the primary ingredients—like flour and sugar—to create the main product. Similarly, neglecting less significant variables allows us to concentrate on the primary factors affecting concentration.

This is a general solution for this equation; the solutions are already there... we club all of them into this constant.

In deriving a general solution for the Gaussian dispersion model, multiple constants associated with dimensions and conditions are combined into a single constant. This streamlining of variables facilitates easier calculations and constructs a cohesive model that adheres to the boundary conditions (i.e., environmental limits). It helps us derive the essential relationships between concentration and time without getting bogged down by the many variables at play.

Think of combining different ingredients in cooking to simplify a dish. Instead of measuring every individual spice every time, a cook may create a blended spice mix that represents a majority of flavors, saving time and ensuring a consistent flavor every time a dish is prepared.

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