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# Box Model for Steady-State Pollution Concentration

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A Power plant is the main point source of SO2 emissions for a 25 km2 area. Estimate the steady-state concentration of SO2 for this area from this point source given the following temperature data. Windspeed= 5km/hr. The concentration of SO2 is 111.1 g/s.

Height (in meters) Temperature (K)
0 300
500 294
1000 288
1500 300

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A Power plant is the main point source of SO2 emissions for a 25 km2 area. Estimate the steady-state concentration of SO2 for this area from this point source given the following temperature data. Windspeed= 5km/hr. The concentration of SO2 is 111.1 g/s.

Height (in meters) Temperature (K)
0 300
500 294
1000 288
1500 300

The increase in temperature after the linear decrease causes an "inversion layer". (http://www.engin.umich.edu/labs/EAST/me589/gallery/CCAQ_f04/mixing_height_inv.htm) explains it well....

"Mixing Height and Inversions
The mixing height is the height of vertical mixing of air and suspended particles above the ground. This height is determined by the observation of the atmospheric temperature profile. A parcel of air rising from the surface of the Earth will rise at a given rate (called the dry-adiabatic lapse rate). As long as the parcel of air is warmer than the ambient temperature, it will continue to rise. However, once it becomes colder than the temperature of the environment, it will slow down and eventually stop. It is at this junction where the temperature of the parcel crosses the curve denoting the vertical environmental temperature profile determines the mixing height.

Inversions are a result of the vertical temperature profile of air. Temperature normally decreases as altitude increases in the troposphere (at an average rate of 1°C per 100 meters). However, an increase of temperature as altitude increases can occur and is called an inversion. Thus, the colder air layer is below the warmer air, resulting in a stable temperature profile that restricts vertical mixing. Because of the restricted mixing volumes of air due to the inversion, pollution becomes stagnant and does not dissipate.

Los Angeles is a prime example of a location that is prone to inversions. Thus, in this location, pollution does not dissipate, causing large amounts of smog and air pollution."

Thus the mixing height for this situation will be at the height where the air begins to get warmer (1000 m).

I found a good, easy to understand, link to the box model as it relates to pollution (http://www.atmos.ucla.edu/AS2/scrns/pdfnotes/05boxmodel.big.pdf)

Assumptions
1. Pollution only travels as far as the inversion layer. This forms the height of the "box".
2. The wind speed is uniform in speed and direction across the entire side of the "box".
3. The "residence time" of pollution particles is governed solely by the wind as it blows across the city.

Area = 25 km2 = 5 km x 5 km = 5000 m x 5000 m

V = volume of the box = 5000 m x 5000 m x 1000 m =2.5 x 1010 m3

S = source rate = 111.1 g s-1

 = residence time (the time that an average particle of pollution spends in the box) = 12500 s

See below for work...
Wind speed = 5 km/hr x (1000m/km) x (1 hr/3600 s) = 1.39 m/s

As stated in the assumptions, the "residence time" of pollution particles is governed solely by the wind as it blows across the city. Look at the diagram above, and think of the wind as a big broom. Each second, air (and pollution) in a 1.39 m x 5000 m x 1000 m volume on the edge of the city gets displaced (moved) from the box. If we follow the journey of a particle as it travels from the middle (an average particle is one halfway into the box or at 2500 m) to the right end of the box, we can calculate the time that this particle will spend in the box.

time = distance = 2500 m = 1800 s
wind speed 1.39 m/s

(I'll take this as the "residence time")

so q = S = 111.1 g s-1 x 1800 s = 8 x 10-6 g m3 (the steady state concentration)
V 2.5 x 1010 m3

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