3 DIMENSIONAL DISPERSION MODEL

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3 DIMENSIONAL DISPERSION MODEL

• Similar to heat conduction equation in 3-d
• Solution for instantaneous release of X g of
  pollutant at t 0 and x y z 0

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2-D Dispersion from a continuous, elevated point
source
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• The concentration profiles in the y and z
  directions look like Gaussian distributions (next
  3 slides)
• By substituting
• we get the standard double Gaussian
  distribution
• Hence called the Gaussian plume equation

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Figure 4-1 Wark Warner

• Gaussian or normal distribution function

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GAUSSIAN (NORMAL) DISTRIBUTION
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DOUBLE GAUSSIAN DISTRIBUTION
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DISPERSION COEFFICIENTS

• Ky and Kz approximately proportional to wind
  speed
• Ky/u and Kz/u approximately constant
• ?y and ?z should vary approximately with
  x(1/2)
• Field observations show more complex variation
  (Figures 6.7 and 6.8 de Nevers)
• Wind speed and solar flux combine to give
  stability classes A - F (Table 6.1 de Nevers)

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• Horizontal dispersion coefficient

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• Vertical dispersion coefficient

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Algebraic expressions for dispersion coefficients

• The graphical representations of ?y and ?z can
  be approximated by algebraic expressions using
  parameters in Tables 4-1 through 4-3 of Wark,
  Warner, and Davis.

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Algebraic representation of dispersion
coefficients Rural environment
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Algebraic representation of dispersion
coefficients Urban environment
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Algebraic representation of dispersion
coefficients Urban environment
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GROUND LEVEL CONCENTRATION ALONG CENTER LINE

• We are most interested in ground level, z0,
  concentrations (where humans and other life forms
  reside),
• On the center line, y0, (where concentrations
  are at their maximum

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2-D STEADY DISPERSION MODEL GROUND
REFLECTION

• From the release height of H above ground,
  dispersion can progress upward towards the mixing
  height. In the downward direction the ground acts
  as a mirror unless the pollutant gets deposited.
• The effect of the ground can be handled
  mathematically by treating the reflection as
  another point source located below ground (at - H)

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Figure 4-3 Wark, Warner Davis

• Use of an imaginary sourceto describe reflection
  at the ground

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Figure 4-4 Wark, Warner Davis

• Effect of ground reflection on pollutant
  concentration

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MAXIMUM GROUND LEVEL CONCENTRATION

• At z 0 (cwith reflection ) 2(cwithout
  reflection )
• Not as simple at other z.
• C(x,0,0) first increases with x due to ground
  reflection but horizontal dispersion (y
  direction) eventually decreases it. (Fig 4-5)
• The location and magnitude of the maximum
  concentration can be determined from the
  equations above. Fig 4-8 provides a convenient
  tool. Other empirical methods are also available

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Figure 4-5 Wark, Warner Davis

• Concentration profiles along the center line of a
  stack plume

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Figure 4-8 Wark, Warner Davis

• Maximum Cu/Q value as a function of stability
  class and downwind distance

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MIXING HEIGHT LIMITATION

• As the ground represents a lower limit to the
  vertical dispersion, the mixing height represents
  an upper limit. Multiple reflections from the
  ground and the stable layer above need to be
  considered giving rise to
• Approximation
• No effect of mixing height for xltxL
• Completely mixed in the z direction for xgt2 xL
• Interpolate (on log-log plot) in between xL
  and 2 xL
• xL corresponds to ?z0.47(L-H)

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ONE DIMENSIONAL SPREADING MIXING HEIGHT
LIMITATION

• After a sufficient distance downstream (say ?z
  mixing height, xgt2xL) ?the plume can only
  disperse horizontally.
• If we consider the plume well mixed in the
  vertical direction, we can obtain
• where L mixing height