Plume Spreading

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Plume Spreading

• Presentation 12

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How a Plume Spreads

• A crucial factor in our ability to deal
  rationally with air pollution is the availability
  of models by which pollution downwind from a
  source can be predicted on a quantitative basis
• One approach has been to argue that the turbulent
  diffusion of pollutants can be treated by an
  average diffusivity constant and an equation
  similar to that in

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How a Plume Spreads

• Such an average "eddy diffusivity constant" would
  be evaluated numerically on the basis of
  empirical studies
• The meandering of a plume is accounted for in
  such models by assuming straight line downwind
  travel of the plume the average lateral
  spreading is governed by a diffusion-like
  equation, analogous to the previous equation, and
  the details of the meandering motion are averaged
  out
• It is generally agreed that such an approach
  envisioning an average diffusive-like behavior
  must incorporate eddy diffusivity constants which
  have no precise significance in the physical
  sense
• Hence sole justification for their use must be
  based on empirical success

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How a Plume Spreads

• Now we shall focus our attention on the lateral
  diffusion of a pollutant from within a plume, and
  using average eddy diffusivity constants, we
  shall derive (yes, derive!) the equation which
  governs the downwind spreading of a plume
• Let ? again denote the concentration of pollutant
  molecules, so that n? is the number of pollutant
  molecules found in a cubic meter of air
• The equation governing the diffusion process can
  be deduced by reference to the following figure

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How a plume Spreads
This illustrates a hypothetical volume,
stationary above ground, through which air and
pollutants move with the wind direction parallel
to the x-axis The dimensions of this volume are
?x, ?y, and ?z, with sides parallel to the axes
of the coordinate system The rate at which the
concentration of a pollutant builds up within a
volume is given by the rate at which pollutant
molecules enter minus the rate at which they leave

• The figure shows diffusion of pollutant molecules
  parallel to the y-axis
• The rate at which pollutants accumulate in the
  volume element is the difference between the rate
  at which they enter and the rate at which they
  leave
• The large arrows show the sense of flow of
  molecules if ??/?y is a negative quantity in the
  region of the volume element

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How a Plume Spreads

• Lets consider only the movement of molecules
  along the y-direction
• By assuming a Fickian type of diffusion, we know
  that the rate at which pollutant molecules
  accumulate within the volume is given solely by
  the difference in the gradients of the
  concentrations at the two faces of the volume
• One face at y and the other at y ?y, with the
  gradient at each face weighted by the eddy
  diffusivity constant at the respective face
• The diffusion of pollutant molecules will be in
  the direction of the arrows shown in the previous
  figure
• If ? decreases with increasing y, there will be a
  net build-up of the average value of ? within the
  volume (if more pollutant molecules enter from
  the left face than leave through the right)
• The rate at which pollutant molecules accumulate
  in a region of the atmosphere of volume ?x?y?z is
  therefore governed by the equation

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How a plume Spreads

• The factor (?x?z) on the right-hand side of this
  equation is the area of the face through which
  the molecules pass
• The subscripts x and z indicate that the gradient
  of the concentration with respect to y must be
  evaluated for a constant value of x and of z
• A subscript y is attached to the diffusivity, D
  because, in a turbulent atmosphere, we must
  anticipate that the diffusivity along the x- and
  z-directions would differ from that along the
  y-direction
• Dividing both sides of the equation by ?x?y?z and
  rearranging, we find

?x?y?z m3 n total molecules / m3 ?
pollutant molecules / total molecules LHS
has units of pollutant molecules / second
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How a Plume Spreads

• The implication of this equation can be seen more
  easily if we assume that the eddy diffusivity is
  independent of position and for the moment drop
  the subscript notation
• The sign of the quantity in curly brackets on the
  right-hand side indicates whether the variation
  of pollutant concentration ? with y has positive
  curvature or negative curvature
• In extreme instances, this corresponds to whether
  the concentration has a local minimum or local
  maximum, respectively, as illustrated in the
  following figure
• This equation predicts that if the concentration
  has a local minimum, then the right-hand side of
  the equation has a positive value, and the
  concentration will increase with time
• If it has a local maximum, the right-hand side
  has a negative value and ? will decrease with
  time
• Thus the above equation properly predicts the
  tendency of the pollutants within a plume to
  spread uniformly throughout the atmosphere along
  the y-direction, orthogonal to the wind

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How a plume Spreads

• An illustration that describes a
  trend toward a uniform distribution of pollutant
  concentration ? in the atmosphere The arrows show
  the direction of diffusion of pollutant molecules
  for (left) a local minimum in ? and (right) a
  local maximum

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How a Plume Spreads

• Now when we turn to consider the diffusion
  process in the vertical z-direction, we conclude
  that it is governed by an equation analogous to
  
• but containing an eddy diffusivity Dz which may
  differ in magnitude from the horizontal
  diffusivity Dy.

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How a Plume Spreads

• This leaves only the movement of molecules along
  the x-direction still to be described
• The introduction and elimination of pollutant
  molecules from the volume for this direction is
  different, because the major cause of the
  movement is the average wind, not the diffusivity
• The rate at which pollutant molecules enter the
  volume is approximately where ?y?z
  is the area of the face which they cross.

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How a Plume Spreads

• The wind speed is so much less than the
  molecular speeds that, when the pollutants enter
  the volume, their concentration readily adjusts
  to the local value ?(x). Similarly, as the wind
  carries air along the x-direction from the face
  of the volume at x to the face at x ?x, the
  relative concentration ? will be affected by the
  number of pollutant molecules which have entered
  or left the volume by diffusion in the y- and
  z-directions
• Thus the concentration at x ? x will in general
  differ from that at x, and the rate at which
  pollutant molecules leave must be written as
• The rate at which they accumulate per unit volume
  is therefore the difference between the rate at
  which they enter and leave

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How a Plume Spreads

• Now if the accumulation of the pollutant
  molecules moving along all three principal
  directions is added together, we find that the
  rate of increase in the relative concentration is
  given by
• This equation describes the accumulation of
  pollutants at any position in the atmosphere

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How a Plume Spreads

• The important point to realize is that a source
  emitting pollutants at a constant rate has a
  plume whose average shape does not change with
  time
• In other words, a steady-state condition
  prevails so the left-hand side of the previous
  equation is zero
• Pollutants are neither accumulated nor lost at
  any position although they continually diffuse
  outward from the center line of the plume (the
  x-axis) the average concentration at any point
  in space remains constant in time
• The equation describing the downwind dispersal
  can be more accurately written if we take the
  limiting form of the previous equation when the
  size of the volume element is considered to be
  infinitesimally small
• In the notation of calculus, we have

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How a Plume Spreads

• where the signs of the type d/dy indicate partial
  derivatives that is, derivatives with respect to
  one variable (in this instance, y) when all other
  independent variables (for example, x and z) are
  considered as constants
• This equation, together with the stipulation that
  ? 0 an infinite distance from the source,
  governs the shape of the plume as pollutants are
  carried downwind
• The eddy diffusion constants Dy and Dz need not
  be equal

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How a Plume SpreadsA Solution

• For the special case in which the diffusion
  constants Dy and Dz in the previous equation are
  independent of position, the type of diffusion
  described by this equation is said to be Fickian
• In this instance, the equation can be solved by a
  simple expression for the concentration ? of the
  pollutant if we also assume that the average wind
  speed u does not vary with position
• Such conditions can be satisfied only
  approximately by actual atmospheric behavior
• If pollutants are emitted from a point source at
  ground level, the solution for is then

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How a Plume SpreadsA Solution

• This equation gives the concentration which would
  exist in the ambient air at any position downwind
  from the source
• We recall that the source is located at the
  origin of our coordinate system (x0, y0,z0),
  and that x is the distance directly downwind from
  the source along the plume line, y is the
  horizontal distance from the plume line, and z is
  the vertical height above the ground

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How a Plume Spreads-A Solution

• If we are interested in what concentration is
  experienced by a receptor, we must evaluate the
  formula by inserting the x-, y-, and
  z-coordinates of the receptor
• Let us examine the significance of this result by
  first turning our attention to the prefactor on
  the right

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How a Plume Spreads-A Solution

• We note that ? is directly proportional to the
  parameter Q, which we use to represent the rate
  at which the source emits the pollutant in
  question.
• The value of Q must be expressed in units
  commensurate with ?, that is, if ? is in
  kilograms per cubic meter, then Q must be what is
  called the mass emission rate Q QM , which is
  expressed as the number of kilograms per second
  of the pollutant issuing from the exhaust stack

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How a Plume Spreads-A Solution

• On the other hand, if ? is to be the fraction of
  air molecules which are pollutant molecules, then
  the emission rate must be the volume emission
  rate Q Qv , which is the volume of the
  pollutant gas emitted each second, evaluated for
  standard pressure (1000 mb) and ambient Kelvin
  temperature T.
• The volume emission rate Qy (in cubic meters per
  second) is related to the mass emission rate QM
  (in kilograms per second) by the equation
• Where MP is the molecular weight of the pollutant
  in kg

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How a Plume Spreads-A Solution

• Therefore, either Qv or QM should be used in
  place of Q in as appropriate.
• gives us an important prediction The profile of
  the plume transverse to the wind is governed by
  what is called a binormal distribution.

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How a Plume SpreadsA Solution

• If for any value of x the receptor is a distance
  y from the plume line, the decrease in ? with
  increase in y follows a normal distribution, as
  described by the following factor contained in
  the solution

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How a Plume SpreadsA Solution

• This is known as an exponential function, where
  the exponent contains the square of the
  independent variable (y)
• The bell-shaped (Gaussian) curve described by
  this factor has a width proportional to the
  parameter ?y called the standard deviation of the
  distribution
• As indicated in the following figure, about 68
  of the pollutant molecules are found within 1
  standard deviation of the plume line, and 96
  within 2 standard deviations
• The concentration at 1 standard deviation is
  about 60 of the concentration on the plume line
  at 2 standard deviations, it is 14

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How a Plume SpreadsA Solution

• As we said previously, the solution is a binormal
  distribution, which means that the decrease of ?
  with vertical distance from the plume line also
  follows a normal distribution
• In general, the diffusion of pollutants
  vertically does not proceed at the same rate as
  it does in the horizontal direction, so the SD ?Z
  usually differs from ?y.
• Since the source is located at ground level, it
  must also be understood that ? 0 for negative
  values of z.
• The Solution as written contains no explicit
  mention of the downwind distance x of the
  receptor, although we certainly expect such a
  dependence
• In fact the x-dependence is contained in the SDs
  ?y and ?Z

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How a Plume SpreadsA Solution

• In order for
  
• to be a solution of
• The SD must satisfy the conditions

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How a Plume SpreadsA Solution

• The square of the standard deviations, known as
  the variances (?2), are thus required to be
  proportional to the respective diffusivities and
  to the downwind distance.
• They are inversely proportional to the wind
  speed.
• From these conditions it is evident that the
  larger the diffusivity for the transverse
  direction, the more rapidly the plume spreads
  with downwind distance.
• The solution given above thus logically accounts
  for a more rapid diffusion of a plume when there
  is a greater intensity of turbulence.
• The shape of the spreading plume as ?y increases
  downwind is illustrated in the following figure
  for the concentration at ground level.

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How a Plume SpreadsA Solution
Source

• The above figure illustrates representative
  ground-level concentration for several transverse
  profiles downwind from a point source.

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How a Plume SpreadsA Solution

• Our preceding discussion assumes that the
  pollutants in the effluent are gaseous, because
  they respond to turbulence and mix just as air
  molecules do.
• The results are not valid for large particles
  which may be in the effluent, because for them
  the sedimentation owing to gravity is an
  important effect.
• In vigorously turbulent areas, particles up to 30
  microns radius do not settle at an appreciable
  rate since they are more responsive to the eddy
  motions and effectively follow the surrounding
  gas, at least for a few hours.
• Thus the diffusion of such small particles can
  also be described by the previous equation
• Larger particles will settle and will produce
  higher concentrations near the source than is
  predicted by this formula

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Random or Correlated

• If the plume width is considered to be the
  standard deviation ? in either transverse
  direction, then
• indicates that the width increases with the
  square root of the distance traveled downwind.
• This feature is characteristic of Fickian
  diffusion from a point source.
• If, on the average, there is a random migration
  of pollutant molecules, then
• is supported by established molecular theory. If
  not, then the problem must be approached with a
  much higher degree of theoretical sophistication.
  
• Any correlation between the behavior of different
  portions of a plume implies that the
  relationships between the variances and distance
  x in the first equation above are invalid.

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Random or Correlated

• Now our discussion on the characteristics of
  atmospheric turbulence shows that in some
  instances there will be a degree of correlation
  with atmospheric diffusion.
• Statistical theories based upon the work of G. I.
  Taylor can relate the parameters ?y and ?z to
  expressions for the correlation of wind
  velocities in different portions of the plume
• However, in most cases we do not know what the
  correlation expressions are for the actual
  turbulence in a given geographical locality.
• Thus we are at a standoff so far as their
  applications are concerned.
• But nevertheless we are warned that for most
  practical cases the Fickian diffusion model is to
  a degree inadequate.
• It is still instructive to consider the general
  features of plume dispersal that the Fickian
  model predicts.

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Random or Correlated

• We might anticipate that these are at least
  qualitatively accurate for describing the
  dispersal in turbulent atmospheres.
• The important features of
• 1. The downwind concentration at any location is
  directly proportional to the emission rate of the
  source.
• This is because Fickian diffusion is a linear
  process, dependent only upon the gradient of the
  concentration.
• 2. The more turbulent the atmosphere, the more
  rapid the spread of the plume in the transverse
  direction.
• Turbulence increases the eddy diffusivity in

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Random or Correlated

• 3. The maximum concentration at ground level is
  found directly downwind, on the plume line, and
  is inversely proportional to the downwind
  distance from the source.
• 4. The maximum concentration decreases for higher
  wind speeds
• Even on the plume line, where at ground level
  there is no explicit dependence on (because
  ?y?z is inversely proportional to ), the
  ground-level concentration will actually decrease
  with increasing wind.
• This is because the eddy diffusivity DZ increases
  with wind speed due to increased mechanical
  turbulence
• These are 4 key features of most models which
  describe the dispersal of emissions from a point
  source at ground level