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 It is also inversely proportional to average wind speed It follows a distribution function that fits normal distribution (Gaussian function)

A More Realistic but Simple Approach Resulting dispersion equation

Eulerian Approach Fixed coordinate system Continuity equation of concentration c i Wind velocities u j consist of 2 components: Deterministic Stochastic

Eulerian Approach u’ j random  c i random  No precise solution Even determination of mean concentration runs into a closure problem

Eulerian Approach Additional assumptions/approximations Chemically inert (R i =0) K theory (or mixing-length theory)  Where K jk 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

Eulerian Approach Solutions An instantaneous source (puff)

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

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 x 1 to x 1 +dx 1, x 2 to x 2 +dx 2, x 3 to x 3 +dx 3

Lagrangian Approach Ensemble of particles. Ensemble mean concentration

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

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

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

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

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

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

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

Session 3, Unit 6 Dispersion Coefficients

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

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

Field Measurements Problem 7.8