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Session 3, Unit 5 Dispersion Modeling

Session 3, Unit 5 Dispersion Modeling. The Box Model. Description and assumption Box model For line source with line strength of Q L Example. A More Realistic but Simple Approach. Basic assumption: Time averaged concentration is proportional to source strength

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Session 3, Unit 5 Dispersion Modeling

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  1. Session 3, Unit 5Dispersion Modeling

  2. The Box Model • Description and assumption • Box model • For line source with line strength of QL • Example

  3. 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)

  4. A More Realistic but Simple Approach • Resulting dispersion equation

  5. Eulerian Approach • Fixed coordinate system • Continuity equation of concentration ci • Wind velocities uj consist of 2 components: • Deterministic • Stochastic

  6. Eulerian Approach • u’j random  ci random  No precise solution • Even determination of mean concentration runs into a closure problem

  7. Eulerian Approach • Additional assumptions/approximations • Chemically inert (Ri=0) • 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

  8. Eulerian Approach • Solutions • An instantaneous source (puff)

  9. 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

  10. 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 x1+dx1, x2 to x2+dx2,x3 to x3+dx3

  11. Lagrangian Approach • Ensemble of particles. • Ensemble mean concentration

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

  13. 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 Eulerian vs. Lagrangian

  14. 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

  15. Physical Picture of Dispersion • Dispersion of a puff under three turbulence condition • Eddies < puff  Significant dilution • Eddies > 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

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

  17. Gaussian Dispersion Model • Ground reflection • Special cases • Ground level receptor (z=0) • Center line (y=0) • Ground level source (H=0)

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

  19. Session 3, Unit 6Dispersion Coefficients

  20. Factors Affecting σ • Wind velocity fluctuation • Friction velocity u* • Monin-Obukhov length L • Coriolis parameter • Mixing height • Convective velocity scale • Surface roughness

  21. 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

  22. Field Measurements • Problem 7.8

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