NUMERICAL SIMULATION OF AIR
POLLUTION DYNAMICS DUE TO
POINT SOURCE EMISSIONS
FROM AN INDUSTRIAL STACK
Erik Minges – UNC Wilmington
April 27, 2010
Outline
•
•
•
•
Statement of the Problem
Background
Hydrodynamic Equations
Discretization Methods
– Finite Difference Method
– Finite Volume Method
• Conclusion
• Works Cited
Statement of the Problem
• Numerical simulation of air pollution from a point
source industrial stack emission based upon the
Navier-Stokes equations in fluid dynamics.
Motives
• Global threat of air pollution
– Major health hazard to humans and ecosystem
– Billions effected, mainly in Eastern nations where
environmental regulations are weak.
– U.N. reported 13 cities as “Brown cloud” hot spots.
– Global weather patterns effected.
– CO2 and other pollutant emissions contributing to
global warming.
Inspiration
• Titan cement plant to open in Castle Hayne.
– Near Cape Fear river basin
• Superfund sites discharge into Atlantic.
• Plant closed 20 years ago because of emissions.
– Many schools and communities few miles from
proposed site. UNCW approximately 10 miles away
– Annual emissions [Friends of the Lower Cape Fear]
• 263 pounds of Mercury (will be 3rd highest mercury
emissions in NC), 348 tons of particulate matter (equivalent
to 22.5 billion cigarettes), 172 pounds of Lead, 6,792 pounds
of Benzene, 21,900 pounds of Ammonia, 4,928 pounds of
Polycyclic Organic Matter
Air Pollution Background
• Natural or man-made substance that causes harm
– Affects health of humans and animals.
– Contaminates soil, damages vegetation, reduces solar
radiation reaching Earths surface, alters weather
patterns, and ultimately effects the global climate
balance.
• About 60% of emissions from point sources
• Major pollutants
– Dust particulates, particulate matter, ozone, sulfur
dioxide, nitrogen oxides, carbon monoxide, and lead.
Conservation Laws
• 1-D conservation laws take on the form
– Rate of change of u = Stuff in – Stuff out
u F (u )

 0, t  0
t
x
u (0, x)  u0 ( x)
– Measurable property of an isolated physical system
does not change as the system evolves.
Solving System of Conservation Laws
• Hydrodynamic equations are a system of nonlinear, hyperbolic conservation laws
– 2D form with source terms
ut  f (u) x  g (u) y  S
– Numerically solve through system of equations
ut  f (u ) x  0


ut  g (u ) y  0 
 u S 0 
 t

Discussion of Mass Conservation

   ( V )  0
t

t
– Time rate of change of the mass density
  V  – Flux of the mass density momentum
• Controls mass entering and leaving system
Discussion of Navier-Stokes equation
( V )
 [( V  )V ]  p
t
Vector equation in x and y
V  V  – Advection term, movement of pollutants
p
– Pressure gradient (path trajectory)
Discussion of Energy Conservation
E
   ( EV )    ( pV )
t
  EV  – Energy flux, amount of energy the system
maintains in a given direction
   pV  – Pressure flux, amount of pressure the
system maintains in a given direction
The Hydrodynamic Equations
• Air pollution dynamics can be described by
– Conservation of mass: 
   ( V )  S   S D
t
– Navier-Stokes equation (Conservation of momentum):
( V )
 [( V  )V ]  p  g  SV
t
– Conservation of energy:
E
   ( EV )    ( pV )  S E
t
The Source Terms
• Sources account for the source of emissions for
pollutants and the interaction between particles.
• Sources provide flux of pollutants into the system,
while conservation laws maintain the fundamental
physical laws that the system must obey.
Description of Source Terms
• Four source terms implemented in code
– Source for diffusion
• Controls molecular diffusion between particles in atmosphere
S D    [ D( x, y, t ,  ) ]
• Assume diffusive coefficient, D is constant to simplify
• Greatly reduces number of degrees of freedom for molecular
diffusion between particles
S D  D 
2
Description of Source Terms
• Emission of pollutants
– Assume width of emitter is x0 and height is y0
– Mass density of pollutants for y  y0
e  e0e

y  y0
x0
, Vey  C
– Source for time-independent emissions
  e 0Vey  y x y0

0


e
,
y

y
0
S   x

0
 0,

y

y
0 

Description of Source Terms
• Source term for momentum of emitted pollutants
^
SV  S Vey y
• Source term for energy of emitted pollutants
pe Vey 1 2
SE 
 V S   S D   V  SV  V  g
  1 x0 2
– Where  
cp
cV
 1.4 .
Numerical Simulation
• Discretization methods are used to numerically
evolve a differential equation over time.
• Two methods used in this project
– Finite difference method
• Derivatives in differential equations replaced by finite
differences that approximate them.
– Finite volume method
• Recast PDE’s into conservation form and discretize them
over small volumes surrounding each point on a meshed
grid.
Finite Difference Method
• Reductionist model for dynamics of air
– Analyze individual particles to interpret how the
evolving system reacts as a whole
• Different ways to approximate derivatives
– First-order methods
1. Forward Difference:
2. Backward Difference:
3. Center Difference:
u uin1  uin

x
x
n
n
u

u
u
i 1
 i
x
x
u uin 1  uin 1

t
2t
Finite Difference Methods
• Need information from past and cell neighbors to
predict future evolution of the system.
Grid structure for first order
finite difference method
Error of First Order Finite Difference
• Use Taylors theorem to derive and find error
1
1
2
f x  x   f ( x)  f ' ( x)x  f ' ' ( x)x  f ' ' ' ( x)x 3  O(x 4 )
2
6
f x  x   f ( x)
1
1
 f ' ( x)  f ' ' ( x)x  f ' ' ' ( x)x 2  O(x 3 )
x
2
6
x  x
f x  x   f ( x)
1
1
  f ' ( x)  f ' ' ( x)x  f ' ' ' ( x)x 2  O(x 3 )
x
2
6
f x  x   f ( x  x)
1
 f ' ( x)  f ' ' ' ( x)x 2  O(x 4 )
2x
6
Stability Criterion for Advection Equation
• Consider backward difference advection equation
n 1
n
n
n

u j  u j  c u j  u j 1 ,   t
x
• Separate time and space components of u nj
– Let u have time-dependent amplitude with
corresponding Fourier-modes that oscillate in space
n i jx
u Q e
n
j
• Amplitude at nth time step – Q n
• Fourier mode at jth coordinate –
– Substituting and simplifying

Q  1  c 1  e
i x

ei jx
Stability Criterion for Advection Equation
• Magnitude: u  Q e
• So,
2
n
j
n
i jx
 Q  1, as n  
n
Q  1  c  cos z   sin z, z   x
2
2
– Look for maximum value to bound Q
   21  c  cos z ( sin z)  2 sin z cos z  0
d Q
dz
2
1  c  cos z  cos z  c  1
t
c
1
x
• CFL condition
Summary of First Order Methods
• Forward / Backward Difference
u nj1  u nj  c u nj1  u nj 
u nj1  u nj  c u nj  u nj1 
– First order accurate in both t and x.
– Difference is one-sided in x.
– Only stable in a single direction.
Summary of First Order Methods
• Center Difference
u
n 1
j
 u c
n
j


u
2
n
j 1
 u nj1

– First order accurate in t and second order in x.
– The difference is centered in x.
– Stable in either direction
Second Order – Leap Frog Method
• More accurate and stable method - Leap Frog
u nj1  u nj1  c u nj1  u nj1 
– Second order accurate in t and x
– The difference is centered in both t and x
– This is the first two-step scheme in time
Finite Difference Method for
Hydrodynamic Equations
• Conservation of mass
Finite Difference for Hydrodynamical
Equations in 2D
• Velocities equation
– x dimension
– y dimension
Simulation Area
Initial Conditions
• Initial Conditions
kg


1
.
29
– Background mass density of air, 00
– Mass density of pollutants, e0  0.100
– Variable horizontal wind velocity
– Initial vertical wind
velocity = 0
m3
Boundary Conditions
• Periodic boundary conditions
– Loop-di-loop boundaries
– Problem at corners
Stability Criterion for Finite Difference
• The stability of this model depends on the
velocity in the y-direction and Δt and Δx.
, for all n.
• Due to the advection term in Euler equation
• When n gets too large
→ Lethal Instability
Density, Velocity, Mass Profiles
SOURCELESS
Results for Full Finite Difference
Finite Volume Discretization Method
• Similar to FDM solutions calculated on meshed
grid, dissimilar calculations over small volume
surrounding each discrete point on meshed grid
• Recast PDE’s into conservative form and then
discretize them
• Utilize conservation of fluxes through control
volume to iterate solutions through time
Discontinuous Solutions
• Conservation law form
ut  f (u) x  0
ut  f u u x  0
• Discontinuous solutions
d b
u ( x, t )dx  f (a, t )  f (b, t )

dt a
• Rankine-Hugonoit equation
   
   
dxs
f x f x

f



u 
dt u xs , t  u x , t

s

s

s
Shocks and Rarefaction Waves
• Burger’s equation: ut  uu x  0
– Non-linear conservation law
dx
– If u=constant, then  u .
dt
Rarefaction wave
Shock Path
• Finite volume method deals with discontinuous
solutions using Godunov and Roe’s method
instead of Rankine-Hugonoit equation.
The Riemann Problem
• Conservation Law Form
U t  F U x  GU y  S
 S  SD 
    u   v 

 u    p  u 2    uv   
0


  
 

 v  t  uv  x  p  v 2  y  g  S Vey 
• Riemann Problem (Discontinuous Solutions)
  L 
u   u 
   L
 v   vL 
    R 
x  0,  u    u R 
 v    vR 
x0
Approximate Conservation Laws
• Isothermal: Wave speed is a function of pressure
– Isothermal atmospheric system
dP
a
d
- The wave speed
• Approximate Conservation Laws:
U t  F (U ) x  0


U t  G (U ) y  0
 U S 0 
t


Godunov’s Method
• Godunov method evaluates approximate
conservation laws by
U
n 1
i

t 
U 
 Fi  1  Fi  1 
x  2
2
n
i
• Inter-cell numerical flux:


F 1  F U 1 0
i
• Stability Criterion:
2
 i 2 
x
n - Maximum wave velocity present throughout
t  n , S max
n
t
the
domain
at
time
.
S max
The Riemann Solver of Roe
• The difference between U solutions is
m
U  U R  U L   
i 1
(F )
i
K
(F )
i
• The corresponding intercell fluxes
F
i
1
2
1
1 m (F ) (F ) (F )
 FL  FR     i i K i
2
2 i 1
Roe’s Method: Isothermal Euler Equations
• Conservation Law: U t  FUU x  0 , U t  GUU y  0
• Jacobian Matrix
 0
F  2
A
 a  u 2
U
  uv
 0
1 0
G 

   uv
2u 0 B 
U
2
2

a

v
v u 

0 1

v u
0 2v 
Eigenvalues and Eigenvectors
• Numerical flux F

(F )
1
 u, 
(F )
2
 u  a, 
(F )
3
ua
– Roe’s matrix eigenvectors for numerical flux:
R(F )
0 1 
 1
 u  a 0 u  a 
 v
1 v 
R (G )
0
1 
 1
  u
 1 u 
v  a 0 v  a 
• Numerical flux G
1(G )  v  a, (2G )  v  a, (3G )  v
Wavestrengths
• Solve for 
m
U  U R  U L    i F  K i( F )
i 1
• A system of 3 equations with 3 unknowns
• Results


a  u   u ( F )
a  u   u
(F )
(F )
1 
,  2  v  v  ,  3 
2a

(G )
1

a  v   v

,
2a
(G )
3

a  v   v

,
2a
2a
(G )
2
 u  u 
Intercell Flux and Godunov’s Method
• Intercell flux expressions
1
1 m (F ) (F ) (F )
F 1  FL  FR     i i K i
i
2
2 i 1
2
1
1 m (G ) (G ) (G )
G 1  GL  GR     i i K i
i
2
2 i 1
2
• Iterate through time steps
U
n 1
i

t 
U 
 Fi  1  Fi  1 
x  2
2
n
i
U
• Stability Criterion: t  x
n
n 1
i
S max

t 
U 
Gi  1  Gi  1 
x  2
2
n
i
y
t  n
S max
Source Terms for Finite Volume
• The source terms take on the same form as before
• Iterate using finite difference method
U in, j 1  U in, j  tS in, j
 S  SD 


S
0

 g  S Vey 


Sourceless Results of FVM
Sourceless Results for FVM
• System shows great stability
• Physical interpretation can be carried out
Full Source FVM
Full Source FVM – High D (.2 m^2/s)
Full Source FVM – Low D (.2 cm^2/s)
Conclusion
• FDM is useful because it is simple to implement,
but there are problems in dealing with
discontinuous solutions.
• FVM was implemented using conservation laws
from which the Godunov and Roe’s methods were
used to deal with discontinuities in the system.
• Trying to model air pollution from an industrial
stack proves FVM are the preferred, most stable,
of the two methods.
Thank YOU!
• My advisor: Dr. Russell Herman
• To all of my honors committee members
– Dr. McNamara
– Dr. Gan
– Lugo
– Bennett
•
•
•
•
Physics and Physical Oceanography department
The Honors Scholars Program
CSURF
And Everyone else…
References
• “Numerical Simulations of Air-Pollutants in Windy
Atmosphere,” Michalczyk and Murawski, Task
Quarterly 6 No. 2, 2002.
• Numerical Methods for Evolutionary Differential
Equations, Ascher, Siam, 2008.
• “Non-Reflecting Boundary Flux Function for Finite
Volume Shallow-Water Models, Sanders, Advances in
Water Resources 25, 2002.
• Finite Volume Methods for Hyperbolic Problems,
Leveque, Cambridge, 2002.
• Riemann Solvers and Numerical Methods for Fluid
Dynamics, Toro, Springer, 2009.