4.2. Experimentally Found Contribution Factor

So, this is the multiple stacks. So, you have several stacks. All of them contributing to this thing, so it is usually additive, but here you are seeing that it is not just additive, it is slightly lower than N raised to 1. What we mean is that the contribution factor by which we multiply centerline concentration from a single stack. So you have multiple stacks in line. So what it means is that the contribution, the additive contribution is not exactly additive, it is found experimentally that it is about N raised to 4 by 5.

In environmental dispersion modeling, we often assume that the contributions of pollutants from different sources add up linearly. For example, if we have two stacks each releasing pollutants, we might think that the total impact would simply be the sum of the impacts from each stack. However, research has shown that this is not always the case. The concept of a 'contribution factor' helps us adjust our calculations. Instead of expecting a direct addition, the contribution from multiple sources is slightly less than the expected total. This is expressed as N^(4/5), which indicates that if we have N sources, their combined contribution to air quality is less than if they all acted independently.

Imagine you have several friends throwing parties at the same time. You might think that each party adds to the total fun of the evening. However, in reality, some guests may choose to split their time between the parties, or they might carpool, which reduces the overall total number of participants at each party. In this analogy, the 'parties' are the sources of pollutants, and the 'guests' are the emissions. The total experience is not just additive because people's choices and behaviors impact how each party (or emission source) operates.

So, number of stacks is not straight additive, it is lesser than that, which means that there is some loss in the process of doing this. It is an experimentally found. You find out that it is not adding, there will be some loss as I said, it does not reach this receptor, it goes somewhere else, maybe there is mixing, it goes up and down.

When we analyze pollution from multiple sources, we realize that not all emitted pollutants reach their intended destination, which is often called a receptor (like a monitoring station). This loss occurs due to various reasons, such as atmospheric mixing or changes in wind direction. For example, if a pollutant is released from a factory stack, it might rise into the atmosphere but then get dispersed over a larger area or even deposited elsewhere. Consequently, the total concentration at any specific receptor point is often lower than what would be expected if the contributions were simply added together.

Consider throwing a handful of confetti into the air. If you throw it straight up, some confetti might drift away due to the wind or fall back down before it reaches a specific point (like the center of a party). Thus, not all of the confetti lands where you intended; some disperses into other areas. Similarly, in pollution modeling, some emissions do not contribute to the pollution levels at a specific monitoring site because they have dispersed too much.

Usually, conservation of mass means that if there is a reduction somewhere, there has to be an addition somewhere else. So which means it is not simply adding. Non-interacting plumes is the assumption which is not true.

In dispersion modeling, one of the key assumptions is that plumes from different emission sources do not interact with each other—meaning, we expect them to behave independently. However, this assumption is flawed. In reality, plumes can overlap and influence each other, leading to complex interactions that are hard to predict. The conservation of mass principle states that the total mass must remain constant, so if some pollutants are lost or mixed differently, their effects may be redistributed rather than just summed up at receptors.

Imagine each plume as a different colored smoke from fireworks. If one firework releases red smoke and another releases blue smoke, the expectation might be that their colors would blend if they meet in the air. However, if the smoke clouds are too dense or turbulent, they might not just mix evenly and can create unexpected color patterns due to varying wind directions and speeds. This unpredictability in behavior highlights the reality that we can’t simply assume that each plume acts independently.

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