How Engineers Calculate the Pollution from SmokeStacks

Smokestacks waft pollution away from ground level and engineers use complex models to design them.
Trevor English

When we look at the world and examine of the pollution producing aspects of it, we might consider smokestacks to be some of the worst offenders. While that is technically true, smokestacks also actually serve an important purpose of keeping ground-level air safe to breathe and help manage the pollution produced. 

Thinking back to the ages of past, pre-industrial revolution, we might romanticize the time period. We might imagine rolling hillsides, the lack of coal fire plants, farming, agriculture, and a down-to-earthiness that we might sometimes yearn for today. While there are certain aspects of that that may be true, we have to remember that this was also a period before proper waste management or air pollution management. 

In pre-industrial days, anytime people burned anything, the smoke would linger near the ground. All of the pollutants, which people were entirely oblivious about, would linger right where people breathed. Even what we'd consider natural pollution, like the particulates released when burning wood for warmth or cooking, can be highly dangerous to the human respiratory system. 

The invention of chimneys

It really wasn't until the 12th century, when chimneys were invented, that humans had any way to manage this pollution and direct it away from their breathing air. When you consider that homes of past were often lit with kerosene lamps, wood, or peat fires, you can get a grasp of just how harmful indoor air pollution would've been back in the day. Definitely not something to romanticize. 

After the invention of chimney, it was gradually iterated on until around about the 18th century, when the Industrial Revolution spread the use of the chimney to industrial buildings like factories and forges.


While 18th century London, Paris, New York – all the large cities of the time – were quite polluted, they would actually have been far more polluted if the manufacturing facilities they hosted didn't use smokestacks. Smokestacks were essential and necessary for taking air pollution and toxic elements that would otherwise settle near the ground, up to higher altitudes so those pollutants could disperse in the atmosphere. Of course, we also now know that high levels of pollutants in the atmosphere can also cause serious problems, so modern smokestacks are also fitted with filters and scrubbers.

Smokestacks ensure that factory emissions are diluted and dispersed. They ensure that the resulting chemicals are dispersed over a wide area where combustion-oriented pollution is present, and thus, the negative effects are partly mitigated in the immediate vicinity. 

Engineers, specifically environmental and civil engineers, have to know how to calculate smokestack emissions in order to design them. If you have a given amount of pollutants coming from a plant, based on geography, location, elevation, and type of pollutant, how tall or wide does the smokestack have to be to effectively disperse the pollutants?

Those are questions engineers have to answer. Before we get to more of the technicalities of that solution, let's first examine why air pollution is so harmful in the first place. In other words, why put in the effort of building smokestacks?

What risk do toxic air pollutants pose?

Air pollution is one of the many unseen dangers of our modern era. Often, it's out of sight out of mind, but it can have dangerous effects on each and every one of us. 

Air pollutants mainly enter the body through respiration, but some materials, such as heavy metals, can settle out of the air and be taken up by plants, and then ingested when the plants are eaten. Other pollutants can be absorbed through the skin.

Once these pollutants enter the body, they can lodge in the lungs or the bloodstream, eventually collecting in various organs, especially the lungs and liver. This, in turn, can lead to serious health problems.

Once the pollutants reach a high enough concentration in your body, the negative health effects can include interference with normal bodily and chemical functions. At the most basic level, the pollutants change the chemical reactions in individual cells, which can cause cell death, cell function impairment, or lead to cancer. 

Air pollutants in this way can cause damage to internal organs, birth defects, and cancer. 

How Engineers Calculate the Pollution from SmokeStacks
Source: San Francisco Department of Emergency Management/Twitter

Most pollutants have some level of a dose-response relationship. In most cases, a little bit of a pollutant over time will not cause serious damage. However, high concentrations of pollutants over a short period of time, or lower but more sustained concentrations over time can cause more serious and long-lasting health problems. However, if you dilute some pollutants enough, the statistical likelihood of them being harmful is reduced, even over prolonged exposure periods. 

In some cases of pollution-dependant cancer, the relationship of pollutant exposure to risk is generally assumed to be linear. In other words, the more you're exposed to a pollutant over time, the more your risk of cancer goes up, all linearly with a direct correlation. 


When it comes to other diseases excluding cancer, the graph is more refined. For diseases other than cancer, there tends to be less risk at lower levels of exposure, but the risk increases exponentially when you get to medium levels of exposure. The risk curve then flattens out somewhat when the exposure gets very high, suggesting that the risk of damage at medium levels and high levels is similar.

Scientists generally calculate a lifetime cancer risk from pollutants by multiplying the maximum lifetime exposure of a pollutant by the dose-response relationship. Utilizing math that tracks along these same lines, environmental engineers and scientists can extrapolate the increased cancer risks in a population living near a certain source, say a large factory. 

In general though, it's important to note that these risk calculations are just estimates. The exact levels of risk and outcome of exposure to a particular pollutant tend to only come to light after examining health and pollution data numbers in actuality. 

In general, engineers and scientists undergoing risk assessments for smokestacks or air pollutants will include a factor of safety into their calculations. 

Calculating smokestack pollution

Now that we've discussed a little about the danger of air pollutants and how engineers might undergo risk assessments for each pollutant, we can get into the technicalities of how engineers determine how large a smokestack needs to be and what effect it would have on the surrounding air.  

When smoke or air exits a smokestack, its behavior depends on any number of variables. These may include the temperature of the smoke relative to the air, wind patterns, volume, size, flow rate, elevation, and more. Because of this, how smoke actually behaves varies dramatically in real-world conditions, but for the sake of simulations and calculations, engineers have categorized smokestack plumes, or the smoke that leaves the stack, into several different plume types. Examples of these can be seen in the image below, from Benoit Cushman-Roisin, of the Thayer School of Engineering at Dartmouth College.

How Engineers Calculate the Pollution from SmokeStacks
Source: Benoit Cushman-Roisin/ Darmouth College

These diagrams should give you a brief introduction to just how hard it is to calculate and measure pollution from any given smokestack. I'm going to have to draw a line in this article with a basic introduction of what smokestacks do instead of running actual calculations of smokestacks. Because, this article isn't strictly academic in nature, but rather one for general education. I'm going to delve into the process of smokestack and plume calculations and stop right before doing anything highly technical. If you want to learn more, an environmental engineering textbook is going to be necessary.

In general, plumes from smokestacks will be Gaussian in flow, or in other words, the concentration of smoke will take on the shape of a bell curve in cross-section. As smoke leaves a stack of a height, h, it will eventually turn into a plume - which is the trail of gas. This plume origination height, H, may be higher than the smokestack height, h, due to the buoyancy of the pollutants. This essentially means the smoke will rise a little bit before it becomes plume. 


While the cross-section of the plume at any given point may take on a variety of shapes, the average cross-section of the plume does in fact end up being close to Gaussian in shape, which is why engineers can generally assume this. 

Take a look at the diagram below to get a little better idea of the concepts we just mentioned. You can clearly see all of the variables in play in one space. 

How Engineers Calculate the Pollution from SmokeStacks
Source: University of Michigan

In general, engineers will be most concerned with the ground level concentration from the plume at a given distance. For example, say there was a town 25 miles down the road from the plume. Environmental engineers would want to calculate the concentration of the plume at a level between the ground and a given height, at a value of x = 25 miles. 

In times of high wind, the pollutants from the smokestack are diluted at higher rates, assuming the same flow rate from the stack. In this case, high winds can help dilute pollutants faster, though wind can also cause turbulence, changing the plume shape and altering the calculations. 

When determining the concentration of a plume at the ground at a given point, the formula might look something like this:

How Engineers Calculate the Pollution from SmokeStacks

Cground being the concentration, S being the emission rate, u being the wind speed at height H, σy being the horizontal dispersion coefficient, σz being the vertical dispersion coefficient, y being cross-wind distance, and H being the effective stack height. 

Without an engineering degree, or at least some understanding of this equation, you probably won't want to attempt the math. This is why I decided to stop this article at a purely introductory point. 

The equation can be resorted and switched around to solve for any given variable, and you could calculate what height you needed your smokestack to be based on an approved concentration of air pollutants at a given distance. Isn't math and engineering just wonderful? It's almost as if there's an empirical data-driven way to know and understand everything around us. Ah, engineering. 

I've really only hit the tip of the iceberg in this post, providing one equation with a few models that underscore the complexities of smokestack design and calculations. I've spared you from Pasquill curves used to determine the dispersion coefficients, from wind velocity profiles based on elevations, and from a variety of other sub-equations needed to make everything balance out. 

Environmental engineers don't just sit around all-day worried about ice melting, they're well-trained professionals working to keep you safe from a wide variety of hazards in the world around us. While you probably can't calculate smokestack equations at this point, hopefully, this post has gotten you to understand the vast complexities of even some of the simplest things around us, even a simple smokestack. 

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