Pollution Dispersion Calculator

Introduction : This calculator calculates the dispersion of pollutants in an unstable boundary layer (convective mixed layer). In this particular case, the boundary layer is statically unstable, with thermals of warm air rising freely from the surface to the top of the mixed layer (zi). Smoke is emitted from a stack of some height (zs), and the wind (M) disperses the pollutant downwind some distance (x). What we want to know is the concentration of the pollutant at this point downwind.

If you are familiar with Gaussian distributions, you should know that this differs slightly from that set of mathematics in that this model accounts somewhat for the vigorous updrafts that occur in this unstable layer, whereas Gaussian models tend to discount these.

In this calculator, you have seven input values:
  1. the source emission rate (Q, grams per second) of pollutant coming out of the stack
  2. the height of the stack (zs, in meters)
  3. the height of the mixed layer (zi, in meters)
  4. the wind speed (M, in meters per second)
  5. the Deardorff velocity (w*, in meters per second). This value is a measure of the velocity of the convective layer, and is a function of the absolute virtual temperature (Tv), the height of the mixed layer, and the effective surface heat flux (FH). In this calculator, we are using a standard value for the Deardorff velocity of 1 m/s. Values are typically between 1 and 2 m/s.
  6. the crosswind-integrated concentration (cy, in grams per meter-squared). This is a measure of the total amount of pollutant with a long box that is crosswind (y direction, if the wind is blowing in an x-direction) to the plume coming from the stack. The box itself is one square meter on each end, with the center of the box at some height.
  7. the distance downwind (x, in meters) of interest. For example, you might want to know the concentration of the pollutant at 1000 meters, 1500 m, etc.

There are five calculated values for this calculator:

  1. the downwind distance of the receptor from the source (X, a dimensionless value).
  2. the source height (Zs, a dimensionless value)
  3. the crosswind-integration concentration (Cy, a dimensionless value)
  4. Concentration (C, a dimensionless value)
  5. the actual concentration of the pollutant at some point x downwind (c, in grams per cubic meter)

The purpose of the calculations of the dimensionless values is that they are used to calculate the plume centerline height and the vertical dispersion distance. These dimensionless values are normalized, that is, they are all set to some value that is relative to all of the values. Once these normalized values have been computed, they can be used to calculate the final value, which is the concentration of the pollutant downwind at some location.

Pollution Dispersion Calculator

Input Values:

Source emission rate (Q, g/s)
Stack top height (zs, m)
Mixed layer depth (zi, m)
Wind speed (M, m/s)
Deardorff Velocity (w*, m/s)
Crosswind-integrated concentration (cy, g/m2)
Distance of a receptor downwind from stack (x, m)

Results:

Downwind distance of receptor from source (X, dimensionless)
Source height (Zs, dimensionless)
Crosswind-integrated concentration (Cy, dimensionless)
Concentration (C, dimensionless)
Concentration of pollutant (c, g/m3)
Concentration of pollutant (c, ug/m3)