Homework

Air pollution dispersion

Box models

1. The simplest of models are single box type and Barry Smith as suggested that the annual mean sulfur dioxide concentration, c microgrammes SO₂ per cubic metre, for a country might be described by the equation:

   

   where e emissions Tg/annum, a the area in square kilometres. What assumptions do you believe are made by this simple model? Zimbabwe emitted 89 ktonne S in 1990 and has an area of 389361 km². What would the model suggest the mean sulfur dioxide concentration to be? Consider the relevance of this value?

2. Single box models are more typically used for cities. The height of the box (h, metres) under stability class D takes the form:

   

   where d is the diameter of the city (in kilometres!). Derive an equation for pollutant concentration at windspeed U and urban emission rate Q. Norwich emits benzene at the rate of 0.23 ktonne/annum. What is the mean concentration likely to be? Multiple box models are a more sensible approach to detailed urban modelling and you will find these often incorporated into commercial models.

Gaussian plume equations

• c, concentration
• q, emission rate
• σ values represent diffusion along the appropriate axes
• y, horizontal distance off plume axis
• z, height
• h, emission height

1. Reduce this equation to the special case where you are interested in ground level concentrations.

2. A smelter emits 5 kg of SO₂ a second at an effective height of 250m where the windspeed is 11m/s under neutral conditions. We wish to determine the ground level concentration 10km directly downwind. [Adapted from Perkins, 1974].

   Determining sigma values

   Sigma values can be determined from charts or formulae once the stability is known. Stability class can be estimated from wind, speed and radiation and are A: Extremely unstable B: Moderately unstable C: Slightly unstable D: Neutral E: Slightly stable F: Moderately stable

   Windspeed at 10m, m/s	Day			Night
   	Incoming solar radiation			Thin cloud or >3/8	< 3/8 cloud
   	Strong	Moderate	Slight
   >2	A	A-B	B
   2-3	A-B	B	C	E	F
   3-5	B	B-C	C	D	E
   5-6	C	C-D	D	D	D
   >6	C	D	D	D	D
   Overcast conditions yeild D both day and night

   Formulae for sigma values, which can be solved using the parameters below from Klug(1969), are:

   and

   Coefficient	Stability Class
   	A	B	C	D	E	F
   Ry	0.469	0.306	0.230	0.219	0.237	0.273
   ry	0.903	0.885	0.855	0.764	0.691	0.594
   Rz	0.017	0.072	0.076	0.140	0.217	0.262
   rz	1.380	1.021	0.879	0.727	0.610	0.500

3. In general calculations may be more difficult than this because you will have to calculate plume rise and wind speed at elevation of height H, compared with the standard measurement height (z), generally 10m:

   

   n=0.5 under stable conditions and 0.25 under unstable conditions, with intermediate values chosen by rough interpolation.

   Determining plume rise

   Plume rise can be estimated from a range of formulae which are desrcibed both in Seinfeld(1986) and Perkins(1974). Estimate plume rise for the Colbert plant of TVA which had 100m stacks of 5m diameter. The flue gas exit velocity is 13.5m/s and the wind speed at 10m is 5m/s. The flue temperature is 145°C and ambient temperature is 15°C. Under unstable conditions the equation is expressed as:

   

   • H_r, plume rise
   • F, bouyancy parameter (it is also possible for plumes to have rise due to momentum)
   • g, acceleration due to gravity 9.8
   • Vs, exit velocity
   • D, stack diameter
   • T_s, stack temperature, K
   • T_∞, ambient temperature, K

4. Plume models are generally derived for instantaneous or continuous releases. However there are situations where there are short term releases. Equations have been derived by Palazzi et al (Atmos.Env. 16, 2785[1982]). A typical one treats the maximum concentration achieved in the puff at substantial distances from the source (ie x > ut/2):

   

   • t, time period of the emission
   • C the concentration calculated on the basis of a gaussian plume
   • σ_x is the diffusion along the axis of advection, may be equated with σ_y
   • u, windspeed
   • the function erf (an error function) may be extracted from tables or rational approximation on a computer.

     A car parks after a long motorway run and emits H2S for at 0.21mg/s for 25 seconds at an effective height of 1m. Determine the maximum H2S concentration 200 m downwind taking u to be 3 m/s and conditions as moderately unstable. Would it be possible to smell the H2S?

References

Resources on plumes