6.2. Fick’s Law and Its Application

Our goal is to predict concentration as a function of place and time (x, y, z, and time). We consider one control volume within the plume, where the pollutant is moving.

This chunk introduces the main objective of the transport model we are studying: to predict the concentration of pollutants in a given area over time. By modeling a specific volume of air where pollutants might exist, we focus on how these pollutants are distributed and transported. The coordinates (x, y, z) help us define the location in three-dimensional space, which, combined with time, provides a comprehensive view of pollutant movement.

Imagine you're in a room where someone is spraying a perfume. Over time, the scent spreads throughout the room. By observing the strength of the scent at different locations (like the corners of the room), we can understand how quickly and in what pattern the perfume disperses. This is similar to how we analyze pollutant concentrations in the air.

We model the box with dimensions delta x, delta y, and delta z, identifying processes such as accumulation, flow in, flow out, dispersion, and reactions.

Here, we are breaking down how we mathematically model the movement and concentration of pollutants. The dimensions of the box (delta x, delta y, delta z) represent the volume we're studying. The term 'accumulation' refers to how much pollutant builds up in this volume versus how much flows in or out and how it spreads due to dispersion. By understanding these processes, we can create an equation that describes how pollutants behave in the environment.

Consider a bathtub filled with water. If you pour water in (flow in) but also have a drain (flow out), the water level (representing concentration) will change depending on how fast you're pouring versus how fast the drain is working. This balance helps us understand how pollutants 'accumulate' in our defined control volume.

The equation for flux, described by Fick’s law, is represented as a negative gradient indicating that pollution tends to move from high to low concentrations.

In this context, we apply Fick's law to describe how pollution disperses. The law states that the flux (the rate of flow of pollutants) is proportional to the concentration gradient. When there's a higher concentration of pollutants in one area, they tend to move towards areas with lower concentrations, which is depicted in the equation as a negative sign. This is fundamental because it helps us predict how pollutants will spread through the air.

Think of a dropped food coloring in water. If you drop it in a corner of a clear glass, the color will spread throughout the water over time, moving from the area of high concentration (where you dropped it) to areas of lower concentration (clear water). Fick’s Law describes this natural tendency of materials to spread out evenly.

We construct our equations by considering the rate of dispersion, with the flow in one direction only. The resulting equation summarizes all the contributing factors.

In this section, we discuss how to write mathematical equations that embody our model of pollutant transport. We assume that pollutants primarily flow in the x-direction and express the changes in concentration over time due to dispersion. The resulting equations help us quantify the impacts of different parameters on pollutant concentration, allowing for predictions about how pollutants will behave under various conditions.

Returning to our bathtub analogy, if we focus only on the water pouring in from one faucet and the drain only on one side, we can create a simple equation to describe the water level's changes. In pollution modeling, we simplify complex real-world inputs to make useful predictions about how pollutants disperse.

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