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1. sufficient heat can be generated by chemical reactions to influence the temperature ... applications of the Lagrangian formulas, the dependence of sy2 and ... – PowerPoint PPT presentation

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Title: Prezentace aplikace PowerPoint


1
Atmospheric Dispersion (AD)
Matus Martini
Seinfeld Pandis Atmospheric Chemistry and
Physics Nov 29, 2007
2
Outline
  • Eulerian approach
  • Lagrangian approach
  • eqns for mean concentration, solutions for
    instantaneous and continuous source
  • Gaussian plume eqn
  • AD parameterizations (P-G curves), plume rise

3
Air pollution dispersion models
  • Box model air pollutants inside the box are
    homogeneously distributed
  • Gaussian model is perhaps the oldest (circa 1936)
  • Lagrangian model - statistics of the trajectories
    of a large number of the pollution plume parcels.
  • Eulerian model - fixed three-dimensional
    Cartesian grid
  • Dense gas model
  • Hybrids (Plume in Grid model)

4
Leonhard Euler (1707-1783) v. Joseph Louis
Lagrange (1736-1813)
5
Lagrange v. Euler
  • PROS over Eulerian models
  • no Courant number restrictions
  • no numerical diffusion/dispersion
  • easily track air parcel histories
  • invertible with respect to time
  • CONS
  • need very large points for statistics
  • inhomogeneous representation of domain
  • convection is poorly represented
  • nonlinear chemistry is problematic
  • Embedding Lagrangian plumes in Eulerian models
    (PinG model)
  • Release puffs from point sources and transport
    them along
  • trajectories, allowing them to gradually dilute
    by turbulent mixing
  • (Gaussian plume) until they reach the Eulerian
    grid size at which
  • point they mix into the gridbox

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8
Eulerian approach
9
Eulerian approach
  • If we assume that the presence of small
    concentration species does not affect the
    meteorology to any detectable measure, the
    continuty eqn can be solved independently of the
    coupled momentum and energy eqns
  • 1. sufficient heat can be generated by chemical
    reactions to influence the temperature
  • 2. absorption, reflection, and scattering of
    radiation by trace gases and particles could
    result in alterations of the fluid behavior
  • The flow of interest is turbulent, the fluid
    velocities uj are random functions of space and
    time
  • deterministic and stochastic component of
    velocity

10
Eulerian approach
  • since uj is random ci resulting from the
    solution must also be random -gt probability
    density function for a random process as complex
    as AD is almost never possible -gt mean of
    ensemble of realizations ltcigt
  • convenient to express ci as ltcigt ci where by
    definition ltcigt 0
  • If the single species decays by a 2nd - order
    reaction
  • ltRgt - k ( ltcgt2 ltc2gt)
  • closure problem (emergence of new dependent
    variables ltc2gt)
  • Eulerian description of turbulent diffusion will
    not permit exact solution even for the mean
    concentration ltcgt

11
Lagrangian approach
  • behavior of representative fluid particles
  • consider a single particle located at location x
    at time t in a turbulent fluid
  • trajectory of the subsequent motion Xx,tt
    at any later time t
  • probability density function
  • integrated over all possible starting points x
  • Q(x, t x, t) transition probability density
    the particle originally at x, t will undergo
    a displacement to x at t

12
Lagrangian approach
  • Ensemble mean concentration
  • General formula for the mean concentration
  • particles present at t0 added from
    sources between t0 and t
  • We need complete knowledge of the turbulence
    properties -gt Q
  • Except the simplest circumstances Q is
    unavailable!
  • Integrals hold only when no undergoing chemical
    reactions, conservative species!

13
  • Eulerian statistics are readily measurable (fixed
    network of anemometers)
  • can include detailed chemical mechanisms
  • serious mathematical obstacle closure problem
  • Lagrangian displacements of groups of particles
    released in the fluid
  • difficulty of accurately determining the required
    particle statistics not directly applicable to
    problems involving nonlin chem reactions
  • Exact solution for the mean concentration even of
    inert species in a turbulent fluid is not
    possible in general!
  • Approximations

14
Mean concentration K theory
  • Eulerian approach approx AD eqn
  • Molecular diffusion is negligible compared with
    turbulent diffusion
  • linearization incompressible atmosphere
  • Ri almost always nonlin, the most obvious approx
  • reaction processes are slow compared w/turbulent
    transport
  • distribution of sources is smooth (violated
    near strong isolated sources)

15
Mean concentration statistical theory
  • Lagrangian approach stationary, homogeneous
    Gausian flow
  • In highly idealized example u(t) is a random
    variable depending only on time, and is
    stationary, Gaussian random proces, u(t) pdf
  • stationarity implies that the statistical
    properties of u at two different times depend
    only on tt and not on t and t individually,
    transition probabilty density
  • Then mean concentration is itself Gaussian (!)

16
Instantaneous point source
  • Eulerian approach
  • eddy diffusivities Kxx, Kyy, Kzz const

Solution
17
Instantaneous point source
  • If we define
  • the two expressions are identical
  • Evidently, there is a connection between Eulerian
    and Lagrangian approaches

18
Continuous source, steady state
  • Lagrangian approach
  • Source began emitting at t 0, mean
    concentration achieves a steady state
    (independent of time), and source strength q in
    g s-1

slender plume approx advection dominates plume
dispersion (neglecting diffusion in the
direction of the mean flow)
19
Continuous point source, steady state
  • Eulerian approach

Solution where
slender plume approx interest only in the plume
centerline
Lagrangian and Eulerian expressions are identical
if
20
Recapitulation
  • Lagrangian and Eulerian solutions are identical
    if
  • Instantaneous point source
  • Continuous point source
  • In most applications of the Lagrangian formulas,
    the dependence of sy2 and sz2 on x are
    determined empirically
  • Relationship between K theory and the Gaussian
    formulas

21
Summary of AD theories, Lagrange / Euler
  • So far only physical processes responsible for
    the dispersion of a cloud or a plume due to only
    velocity fluctuations (instantaneous or
    continuous source in idealized stationary,
    homogeneous turbulence)
  • Because of the inherently random character of
    atmospheric motions, one can never predict with
    certainity the distribution of concentration of
    marked particles emitted from a source. Although
    the basic equations describing turbulent
    diffusion are available, there does not exist a
    single mathematical model that can be used as a
    practical means of computing atmospheric
    concentrations over all ranges of conditions.
  • The deciding factor in judging the validity of a
    theory for atmospheric diffusion is the
    comparison of its predictions with experimental
    data. Theory gives ensemble mean concentration
    ltcgt, whereas a single experimental observation
    constitues only one sample from the
    hypothetically infinite ensemble of observations.
    (Its practically impossible to repeat an
    experiment more than a few times under identical
    conditions in the atmosphere.)

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23
Gaussian spreading in 2D have a binormal
distribution
24
Plume rise Dh
H effective stack height
25
Gaussian Plume Equation
  • Lagrangian approach under certain idealized
    conditions (stationary, homogeneous turbulence),
    the mean conc. of species emitted from a point
    source has a Gaussian distribution
  • in the slender plume case sx -gt 0

f - crosswind dispersion g vertical dispersion
g1 no reflections, g2 reflection from the
ground, g3 - reflection from an inversion aloft
q emission rate, H effective stack height
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Gaussian Plume Equation
  • Eulerian approach It can be shown (use of Green
    function) that we can get to the same result by
    solving the AD eqn (but with const eddy
    diffusivities)!

Johann Carl Friedrich Gauss (1777-1855)
28
Dispersion parameters in Gaussian models
  • Derived from concentrations measured in actual
    atmospheric diffusion experiments
  • where sv , sw are standard deviations of the
    wind velocity fluctuations
  • Fy , Fz characterize PBL
  • friction velocity u, convective velocity scale
    w
  • Monin-Obukhov length L
  • Coriolis parameter f
  • mixed layer depth zi (upper boundary, the height
    of an elevated layer impermeable to diffusion)
  • surface roughness z0
  • height of pollutant release above the ground H

29
AD Parameterizations
  • From two standard deviations more is known about
    sy , since most experiments are ground-level.
    Vertical concentration distributions are needed
    to determine sz
  • Ground-leveled releases are not exactly gaussian
    in vertical.
  • For complete parameterization we need all these
    variables
  • not always available!
  • Pasquill stability categories A F (1961)
  • Surface windspeed
  • Daytime incoming solar radiation
  • Nighttime cloud cover
  • Correlations for sigmas based on readily
    available ambient data!

30
AD Parameterizations
Pasquill stability classes
31
Pasquill-Gifford (P-G) curves
Horizontal and
vertical dispersion coeff
Distance from source m
32
Behavior of a plume
  • initial source conditions exit velocity,Tplume
    Tair
  • stratification
  • wind speed
  • gases are usually released at T hotter than the
    ambient air and are emitted with considerable
    initial momentum
  • Buoyant plume Initial buoyancy gtgt initial
    momentum
  • Forced plume Initial buoyancy initial
    momentum
  • Jet Initial buoyancy ltlt initial momentum

33
Analytical properties of Gaussian Plume Eqn
  • along the centerline (y0) at the ground
    (z0)
  • we need effective stack height H !!
  • Maximum ground-level concentration
  • derivative w.r.t x 0
  • critical downwind distance xc , critical wind
    speed uc

34
Critical downwind distance as a function of
source height and a stability class(plume that
has reached its final height)
no xc for stable stratification!
35
Summary 3
  • Gaussian expressions
  • fail near the surface, since no vertical shear
    is present
  • no chemical reactions, either
  • Eulerian approach
  • AD eqn provides more general approach (special
    cases uniform wind speed and constant eddy
    diffusivities), key problem is to choose the
    functional forms of the wind speeds and the eddy
    diffusivities
  • Generally, exact solution for the mean
    concentration even of inert species in a
    turbulent fluid is not possible!
  • Therefore approximations, K-theory,
    linearizations
  • in stationary, homogeneous Gausian flow the
    solution for ltcgt is itself Gaussian!
  • Instantaneous and continuous point source
    stationary, homogeneous turbulence, and const
    eddy diffusivities -gt Gaussian plume eqn
    (Lagrange agrees with Euler)
  • Experimental data gt parameterizations, P-G
    curves convenient for determining sy , sz
  • We saw why the stack height and PBL meteorology
    matter.
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