Session 3, Unit 5

Dispersion Modeling

The Box Model

Description and assumptionBox model

For line source with line strength of QL

Example

Au

MC AA

WHu

LQC LA

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)

QC

uC

1

2

2

2

)(exp

2

1)(

ax

xp

A More Realistic but Simple Approach

Resulting dispersion equation

22

2

2

2

2

2

1exp

2

2

)(exp

2

1

2exp

2

1

zyzy

zzyy

zzy

u

QC

Or

zzy

u

QC

Eulerian Approach

Fixed coordinate systemContinuity equation of concentration ci

Wind velocities uj consist of 2 components: Deterministic Stochastic

),(),,,( 1

2

tSTccRxx

cDcu

xt

ciNi

jj

iiij

j

i x

'jjj uuu

Eulerian Approach

u’j random ci random No precise solutionEven determination of mean concentration runs into a closure problem

Eulerian ApproachAdditional 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

k k

jkj x

cKcu ''

0

j

j

x

u

),( tSx

cK

xx

cu

t

c

jjj

jj

j x

Eulerian ApproachSolutions An instantaneous source (puff)

tK

z

tK

y

tK

tux

KKKt

Stzyxc

zzyyxxzzyyxx

444

)(exp

)()(8),,,(

222

21

23

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

zzyyzzyy

K

z

K

y

x

u

xKK

qzyxc

22

21 4exp

)(4),,(

Lagrangian ApproachConcentration 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

1),(

),(),,,( 321321

xx

xx

dt

dtdxdxdxtxxx

Lagrangian Approach

Ensemble of particles. Ensemble mean concentration

m

ii ttc

1

),(),( xx

Lagrangian Approach

Solutions Instantaneous point source of unit

strength at its origin, mean flow only in x direction

Continuous source

)(2)(2)(2

)(exp)()()()2(

1),,,(

2

2

2

2

2

2

23 t

z

t

y

t

tux

ttttzyxc

zyxzyx

2

2

2

2

22exp

2),,(

zyzy

zy

u

qzyxc

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

tKtKtK zzzyyyxxx 2,2,2 222

u

xK

u

xKzz

zzyy

y

2,

222

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. continuousDescription of plumeTime 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

)(2)(2)(2

)(exp)()()()2(

),,,(2

2

2

2

2

2

23 t

z

t

y

t

tux

ttt

Qtzyxc

zyxzyx

2

2

2

2

2

2

2

2

2

)(exp

2exp

2),,(

2

)(

2exp

2),,(

zyzy

zyzy

Hzy

u

qzyxcOr

Hzy

u

qzyxc

Gaussian Dispersion Model

Ground reflection

Special cases Ground level receptor (z=0) Center line (y=0) Ground level source (H=0)

2

2

2

2

2

2

2

)(exp

2

)(exp

2exp

2),,(

zzyzy

HzHzy

u

qzyxc

Gaussian Dispersion Model

Maximum ground level concentration and its location Graphical solution

Accuracy of the Gaussian dispersion model

2

2

2exp

1

zzy

H

q

uc

Session 3, Unit 6

Dispersion Coefficients

Factors Affecting σ

Wind velocity fluctuationFriction velocity u*

Monin-Obukhov length LCoriolis parameterMixing heightConvective velocity scaleSurface roughness

Pasquill-Gifford Curves

Condense all above factors into 2 variables – stability class and downwind distanceChartsNumeric formulasAveraging time 3-10 minutes EPA specifies 1 hour

Field Measurements

Problem 7.8