Session 3, Unit 5 Dispersion Modeling

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Session 3, Unit 5Dispersion Modeling
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The Box Model

• Description and assumption
• Box model
• For line source with line strength of QL
• Example

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A More Realistic but Simple Approach

• Basic assumption
• Time averaged concentration is proportional to
  source strength
• It is also inversely proportional to average wind
  speed
• It follows a distribution function that fits
  normal distribution (Gaussian function)

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A More Realistic but Simple Approach

• Resulting dispersion equation

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Eulerian Approach

• Fixed coordinate system
• Continuity equation of concentration ci
• Wind velocities uj consist of 2 components
• Deterministic
• Stochastic

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Eulerian Approach

• uj random ? ci random ? No precise solution
• Even determination of mean concentration runs
  into a closure problem

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Eulerian Approach

• Additional assumptions/approximations
• Chemically inert (Ri0)
• K theory (or mixing-length theory)
• Where Kjk is the eddy diffusivity, and is
  function of location and time
• Molecular diffusion is negligible
• The atmosphere is incompressible
• Resulting semiempirical equation of atmospheric
  dispersion

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Eulerian Approach

• Solutions
• An instantaneous source (puff)

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Eulerian Approach

• A continuous source
• Plume is comprised of many puffs each of whose
  concentration distribution is sharply peaked
  about its centroid at all travel distances
• Slender plume approximation the spread of each
  puff is small compared to the downwind distance
  it has traveled
• Solution

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Lagrangian Approach

• Concentration changes are described relative to
  the moving fluid.
• A single particle
• A single particle which is at location x at time
  t in a turbulent fluid.
• Follow the trajectory of the particle, i.e. its
  position at any later time.
• Probability that particle at time t will be in
  volume element of x1 to x1dx1, x2 to x2dx2, x3
  to x3dx3

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Lagrangian Approach

• Ensemble of particles.
• Ensemble mean concentration

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Lagrangian Approach

• Solutions
• Instantaneous point source of unit strength at
  its origin, mean flow only in x direction
• Continuous source

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Eulerian vs. Lagrangian

• Eulerian
• Fixed coordinate
• Focus on the statistical properties of fluid
  velocities
• Eulerian statistics are readily measurable
• Directly applicable when there are chemical
  reactions
• Closure problem no generally valid solutions

• Lagrangian
• Moving coordinate
• Focus on the statistical properties of the
  displacements of groups of particles
• No closure problem
• Difficult to accurately determine the required
  particle statistics
• Not directly applicable to problems involving
  nonlinear chemical reactions

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Eulerian vs. Lagrangian

• Reconcile the solutions from the two approaches
• Instantaneous sources
• Continuous sources
• Limitation for both approaches
• Lack of exact solutions
• Solutions only for idealized stationary (steady
  state), homogeneous turbulence
• Rely on experimental validation

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Physical Picture of Dispersion

• Dispersion of a puff under three turbulence
  condition
• Eddies lt puff ? Significant dilution
• Eddies gt puff ? Limited dilution
• Eddies puff ? Dispersed and distorted
• Molecular diffusion vs. atmospheric dispersion
  (eddy diffusion)
• Instantaneous vs. continuous
• Description of plume
• Time averaged concentrations for continuous
  sources

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Gaussian Dispersion Model

• Same as Lagrangian solutions
• For an instantaneous sources (a puff)
• For a continuous source at a release height of H

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Gaussian Dispersion Model

• Ground reflection
• Special cases
• Ground level receptor (z0)
• Center line (y0)
• Ground level source (H0)

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Gaussian Dispersion Model

• Maximum ground level concentration and its
  location
• Graphical solution
• Accuracy of the Gaussian dispersion model

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Session 3, Unit 6Dispersion Coefficients
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Factors Affecting s

• Wind velocity fluctuation
• Friction velocity u
• Monin-Obukhov length L
• Coriolis parameter
• Mixing height
• Convective velocity scale
• Surface roughness

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Pasquill-Gifford Curves

• Condense all above factors into 2 variables
  stability class and downwind distance
• Charts
• Numeric formulas
• Averaging time
• 3-10 minutes
• EPA specifies 1 hour

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Field Measurements

• Problem 7.8