Gaussian plume dispersion model
Derivation of the Gaussian plume model
Distribution of pollutant concentration c in the flow field (velocity vector u ≡ ux, uy, uz)
in PBL can be generally described by Reynolds equation in the form:
c
  c    c    c 
    z
 (u )c    x
    y

t
x  x  y  y  z  z 
The inercial or advection term is
non-linear - it contains the product
of unknown parameters u and c:
(u )c  u x
c
c
c
 uy
 uz
x
y
z
The Reynolds equation can be
solved only numerically (along with
other three equations describing
the components of the flow velocity
vector).
x ,  y , z
are turbulent diffusion
coefficients for each spatial
directions
Derivation of the Gaussian plume model
Gaussian model is simplified - does not include a transport in complex flow fields.
The model describes only smoke plume drift by constant velocity along a linear path.
Derivation of this model, therefore, does not take into account the nonlinear
advection term and based on equation involving only the turbulent dispersion.
c
  c    c    c 
   z
 (u )c    x
   y

t
x  x  y  y  z  z 
c   c    c    c 
   z
  x
   y

t x  x  y  y  z  z 
For constant and isotropic turbulent dispersion
simply in the form:
c
 c
t
x   y  z   is the equation
 2c  2c  2c
c  2  2  2
x
y
z
This simplified spherically symmetric diffusion equation has an analytical solution in the
form of a concentration function of two variables (t, r)
 r2 
Q

c (r , t ) 
exp 
32
4

t
8t 


Q is mass flow of pollutant (g/s)
r is distance from the centre (from
source)
Derivation of the Gaussian plume model
Analytical solutions - the spherically symmetric function of time and distance from
the source is expressed by the graph:
It is obvious that the concentration is high but
rapidly decreasing with increasing distance
from the center in a short time t1 after the
pulse emission of pollutant from the source.
Concentration after a longer time t2 is lower
with slow decline.
To derive the model of plume is necessary to establish transport - a shift from the source
in the direction of the x-axis. We can simple define:

r  x  u xt   y 2  z 2
2

12
ux is constant velocity of drifting flow
Derivation of the Gaussian plume model
We have the function of pollutant concentration including the shift in the x-axis:
 x  u xt 2  y 2  z 2 
Q

c (r , t ) 
exp 
32

4t
8t 


u x t3
t1
t2
t3
t1
t2
t3
Plume model is obtained by integrating over all states in the time t  


 x  u xt 2  y 2  z 2 
 ux y 2  z 2 
Q
Q
dt 
c ( x, y, z )  
exp 
exp

32

4

t
4

x
4

x


8

t


0



Derivation of the Gaussian plume model
Resulting formula after integration (and neglect of small terms) expresses the timeindependent 3D field of spatial distribution of pollutant concentration corresponding
to smoke plume.


 ux y 2  z 2 
c ( x, y, z ) 
exp
4x
4x 

Q
The anisotropy of turbulent dispersion can be re-introduced for the y and z-axes:
   yz
 ux y 2 
 ux z 2 
 exp 
c ( x, y , z ) 
exp 
 4 x 


4

x
4  y  z x
y 
z 


Q
The relations for dispersion parameters corresponding to Gaussian standard
deviations are finally introduced.
y 
2 y x
ux
, z 
2 z x
ux
Derivation of the Gaussian plume model
The resulting formula of Gaussian smoke plume model is:
 y2 
 z2 
Q
c ( x, y , z ) 
exp  2  exp  2 
 2 
2 u x y z
y 
 2 z 

The fraction in the formula
expresses the concentration in the
plume axis. By dimensions is:
Q
mg s

 m g m3
2 u x y z m s  m  m
The exponential terms represent a
lateral dispersion in the y and zaxes, they are dimensionless and
can have values ​in the interval (0,1)
Application of the Gaussian plume model
In practical applications, z does not mean the distance from the axis of plume but
represents the height difference between the ground level at the source and at the
reference point (point where the pollutant concentration is calculated).
 y 2    z  h 2 
 z  h 2 
  exp 

c ( x, y , z ) 
exp  2 .exp 
2
2




2 y z u
2 z 
2 z 

 2 y   
Q
The reflection of pollutant from the ground at the level of source is also included in
the formula.
h - is effective height of
the source (stack).
Application of the Gaussian plume model
The second possibility is the inclusion of pollutant reflection from the ground level at
the reference point.
This approach is appropriate to calculate the concentration of the pollutant in
reference points on the plateau.
 y2 
 z  h 2 

c ( x, y , z ) 
exp  2 . exp 
2


 y z u
2 z 

 2 y 
Q
.
Application of the Gaussian plume model
Both approaches for calculating of pollutant reflection from the ground can be
included continuously by a factor theta.
  z  h2 
 y 2 
 z  h2 
Q

  1    exp 
c
 exp  2 1    exp 
2
2




2.   y   z  u
2 z 
2 z 

 2 y 


This factor is calculated as
the ratio of vertical surfaces
defined at the junction
between the source and the
reference point as shown.

A1
A2
Theta factor can have
values ​in the interval (0,1)
A2 (rectangle)
A1
Reference
point
Application of the Gaussian plume model
The dispersion parameters  y ,  z are dependent on the length of the plume xaxis. They are calculated based on the stability classification of the atmosphere by
the tabulated parameters ay, az and by, bz.
 y  ay x
by
 z  az xb
z