Mathematics of Air: Modelling of Pollutant Transport in ...humboldtcanada.com/presentations_air/stockie.pdf · Most industry-standard software is based on Gaussian plume solutions

Background: Atmospheric DispersionThe Gaussian Plume Solution

Application to Trail Smelter

Mathematics of Air:

Modelling of Pollutant Transportin the Atmosphere

John Stockie

Department of MathematicsSimon Fraser University

http://www.math.sfu.ca/˜stockie

May 13, 2011

Humboldt Symposium on “Air”

Mathematics of Pollutant Transport in Air John Stockie – SFU 1/41



Acknowledgments

Enkeleida Lushi

MSc student intern

Funded jointly by MITACS and Teck Cominco

Evgeniy Lebed

Summer undergraduate research assistant

Funded by NSERC

Ed Kniel and Rob Fogal

Environmental Management Group




Acknowledgments

Portions of this work were completed during 2007 while I was aHumboldt Fellow at:

Fraunhofer Institut Techno- undWirtschaftsmathematik, Kaiserslautern

(Institute for Industrial Mathematics)




Motivation

Teck Cominco operates one of the world’s largest integratedlead-zinc smelting operations in Trail, British Columbia, onthe Columbia River.

It is a major driver of the BCeconomy and one of Canada’sbiggest polluters (lead,arsenic, cadmium, mercury,zinc, SO2, . . . ).

Annual emissions reporting isrequired under the NationalPollutant Release Inventoryhttp://www.ec.gc.ca).


http://www.ec.gc.ca



Historical Sidebar

Trail was founded in 1890s as amine supply point. A small smelterwas built.

In 1906, Cominco was formed.

In 1941, WA state was awardeddamages from Cominco fortrans-border pollution – one of themost-cited international law cases.

From 1917 to 1940, emissions ofSO2 were 100–700 T/yr. Currentlydown to 22 T/yr.

Today Teck Cominco prides itselfon its “clean” Trail operations.




The Problem

Teck’s Trail operation is currentlyunable to directly measure Znstack emissions in a reliable orcost-effective manner.

Past Zn emissions reporting reliedon simple “engineering estimates”of chemical processes.

Measurements of the followingparticulates were made using“dustfall jars” during 2001–2002:

zinc (Zn), sulphates (x-SO4),strontium (Sr).




Key Questions

1 Is there a robust method that will provide reliable estimates ofstack emission rates based on deposition measurements?

2 And what is “reliable”? . . . errors of 25-50% in estimates areconsidered acceptable, as long as they are overestimates!




My Aims

To take you on a mathematical journey through . . .

Models: illustrating a model of a physical phenomenon(here, model = equations).

Applications: demonstrating how “classical” mathematicscan be applied to a problem of direct importance to industry.

Generalizations: focusing on underlying generalmathematical structures.

To a mathematician, the beauty of a problem is in its structure!




Outline

1 Background: Atmospheric Dispersion

2 The Gaussian Plume Solution

3 Application to Trail Smelter




Atmospheric Dispersion

Atmospheric dispersion refers to transport of airbornecontaminants via two processes:

1 advection by the wind, and2 turbulent diffusion.

Reduces to solving the advection-diffusion equation

∂C

∂t+ ~u · ∇C = ∇ · (K∇C ) + Q

whereC (~x , t) = unknown contaminant concentration (kg/m3)~u(~x , t) = given wind velocity (m/s)

K = turbulent eddy diffusivity (m2/s)Q(~x , t) = emission source term (kg/m3 s)




Atmospheric Dispersion

A variety of approaches have been used for the solving theadvection-diffusion equation:

analytical – Fourier series, Green’s functions, Laplacetransforms, asymptotics.computational – finite difference, finite volume, spectral.

Most industry-standard software is based on Gaussian plumesolutions (see “Recommended Models” at www.epa.gov).

Most previous work has focused on the forward problem:

Given a set of source emission rates, calculate deposition

and much less on the inverse problem:

Given a set of deposition values, calculate emission rates


http://www.epa.gov



Other Applications




Other Applications




Other Applications

Insect infestations: locusts, mountain pine beetles.Radioactive particles (e.g., Fukushima reactor accident).Ash from volcanic eruptions.Dust and exhaust from automobiles.Contaminant transport in groundwater.Motion of pedestrians in crowds.Diffusion of languages/dialects in populations. Etc., etc. . .

Contaminant transport in groundwater Mount Merapi eruption, Java (2010)




Outline







Simplifying Assumptions

Typical assumptions for a single, isolated stack:

Stack is an elevated point source located at (0, 0,H).

Constant emission rate Q.

Wind velocity ~u = (U, 0,Wset) is constant with Wset � U.

Ground-level deposition is at constant speed Wdep.

Diffusion downwind is negligible compared to convection(Kx = 0).

Steady state – we’ll relax this later.




Governing Equations

Advection-diffusion equation reduces to:

U∂C

∂x︸︷︷︸wind

−Wset∂C

∂z︸︷︷︸settling

=∂

∂y

(Ky

∂C

∂y

)+

∂

∂z

(Kz

∂C

∂z

)︸︷︷︸

cross-wind and vertical diffusion

+Q δ(x) δ(y) δ(z − H)︸︷︷︸point source

Boundaryconditions:

Kz∂C

∂z+ WsetC = WdepC at z = 0 (deposition)

C → 0 as x , y → ±∞ and z →∞

y

wind

effectiveheight

speed

plume centerline

U

z=H

z=0x

z




Gaussian Plume Solution

Assume no settling or deposition (Wset = Wdep = 0) and useLaplace transforms to obtain the Gaussian plume (GP) solution:

C (x , y , z) =Q

2πUσyσzexp

(− y2

2σ2y

) [exp

(− (z − H)2

2σ2z

)+ exp

(− (z + H)2

2σ2z

) ]where σ2

y ,z(x) = 2xKy ,z/U.




Typical Solution Contours

Ground!level concentration

x

y

0 500 1000 1500 2000!250

!200

!150

!100

!50

0

50

100

150

200

0.5

1

1.5

2

x 10!6

1e!06

3.16e!06

1e!053.16e!050.0001

Concentration on plume centerline

xz

0 100 200 300 4000

10

20

30

40

50

Note the peak in ground-level concentrationdownwind of the elevated source.




Examples: Plume Animations

Actual plume dynamics are more complex:1 Numerical simulation of a stack plume.2 Eyjafjallajokull volcanic eruption, Iceland.3 Cesium-137 emissions from Chernobyl, Ukraine.




Ermak’s Solution with Deposition and Settling

Ermak (1977) derived a modified GP solution with deposition andsettling:

C(x , y , z) =Q

2πUσyσzexp

„− y 2

2σ2y

«exp

„−Wset (z − H)

2Kz− W 2

set σ2z

8K 2z

«×

"exp

„− (z − H)2

2σ2z

«+ exp

„− (z + H)2

2σ2z

«

−√

2πWo σz

Kzexp

„Wo(z + H)

Kz+

W 2o σ2

z

2K 2z

«erfc

„Wo σz√

2 Kz

+z + H√

2 σz

«#

where Wo = Wdep − 12Wset.

Key: Deposition flux is linear in Q!! That is, WdepC∣∣z=0

= PQ.




Outline







What’s Left?

So far, we’ve considered a single source in a constant wind.

We still need to include:

Time-dependent wind velocity, not aligned with x-axis:

Shift and rotate coordinatesSum over discrete time intervals to get total deposition

Multiple sources and receptors

Multiple contaminants (Zn, SO4, Sr)

Ultimately . . . solve the inverse problem




Multiple Sources

Four sources (Sn) and nine “dustfall jars” or receptors (Rn):

400 metresScale:

100 2000 300

R8R9

S3

S4

R6R5

R4

R3

R1 S1R2

S2

R7




Multiple Sources

Sources Qs with s = 1, 2, 3, 4.

Total deposition zinc D from all sources is

D =4∑

s=1

PsQs

where Ps depends on the location, wind, and problemparameters.




Sample Output

One month cumulative zinc deposition using actual wind data:

0.05

0.05

0.05

0.05

0.05

0.05

0.05

0.05

0.1

0.1

0.1

0.1

0.1

0.1

0.1

0.1

0.1

0.1

0.5

0.50.5

0.5

0.51

11

x (m)

y (m

)

Contours of Zn concentration (g/m2)

S1 S2

S3

S4

R1 R2

R3 R4

R5

R6

R7

R8

R9

0 500 1000!500

0

500

1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Wind histogram

500

1000

30

210

60

240

90

270

120

300

150

330

180 0

Due to Columbia Rivervalley, wind isunidirectional!




Multiple Receptors

Receptor measurements Dr with r = 1, 2, . . . , 9.

Total deposition of zinc from all sources is

Dr =4∑

s=1

PrsQs

or in matrix-vector form

~D = P~Q

where~D is a 9-vector~Q is a 4-vectorP is a 9× 4 matrix




Inverse Problem

Equations ~D = P~Q represent an over-determined system forthe Q’s, consisting of 9 equations in 4 unknowns.

We actually have three such systems for zinc, sulphate,strontium:

~DZn = PZn~QZn, ~DSO4 = PSO4 ~QSO4 , ~DSr = PSr ~QSr

=⇒ 27 equations in 12 unknowns.

Additional linear constraints on ~Q’s arise from chemicalprocesses.

Conclude: This can be solved as a constrained linear least squaresproblem (Gauss again).




Ill-conditioning

The ill-conditioned nature of this inverse problem iswell-studied in a series of papers by Enting & Newsam (1988).

When measurements are taken at very long range or at highaltitudes, then ill-conditioning can be severe.

BUT . . . “The relatively mild degree of ill-posedness in the[close-range] surface source deduction problem makes thenumerical inversions feasible.”




Numerical Simulations

Use Matlab’s constrained linear least squares solver lsqlin.

Each run requires approx. 30 sec on a Mac laptop – fast!

Use “engineering estimates” of zinc emissions as a guide:

~QZn ≈ [35, 80, 5, 5] tons/yr




Typical Deposition Measurements

Zn and SO4 data are mostly consistent from month to month.

Exception: Zn depositions at R3.

Sr data exhibit more variation (experimental error?).




Results: Emission Rates

Measured depositions Computed emission rates

=⇒




Results: Emission Rates

Measured depositions Computed emission rates

=⇒




Conclusions (so far)

The method seems to captures total Zn emissions well(GP approach conserves mass approximately).

Individual Zn emission estimates still vary considerably.

Hypothesis: Discrepancies in the inverse solution can beattributed to measurement errors, specifically at R3.




Results: Without R3

Emission estimates without R3 are much more consistent!




What’s the Problem with R3?

There is a debris pile containing zinc tailings adjacent to R3!




Summary

Applied the Gaussian plume model to contaminant transport.

Linearity in Q allowed superposition of all sources.

Emission rates can be estimated from measured depositiondata using linear least squares method.

Consistent estimates are found for total zinc emissions . . .which is all that’s required from a regulatory standpoint!

Teck Cominco incorporated our results into their annualreporting to Environment Canada.

These results are published in:

E. Lushi & JMS, Atmospheric Environment,44(8):1097–1107, 2010

JMS, SIAM Review, 53(2):349–372, 2011




Summary: Mathematical Structure

The same approach can be used for a diverse collection ofphenomena involving transport of contaminants, pollen, dust,groundwater, crowds, languages, etc.

The common thread of is the underlying mathematicalstructure:

partial differential equations (advection + diffusion)over-determined linear systemsill-conditioningconstrained optimization (least squares)




Current and Future Work

We recommended that:

A full year’s wind and deposition data be collected.More receptors be added.

Teck Cominco has just completed another round ofmeasurements (May 2010–Feb. 2011) which I aminvestigating right now with . . .

Sudeshna Ghosh

PhD student

Internship funded jointly byMITACS and Teck Cominco

We also plan to validate our results using direct simulations ofthe advection-diffusion equation (“forward problem”).




Thank-you!




References I

Donald L. Ermak.

An analytical model for air pollutant transport and deposition from apoint source.

Atmos. Env., 11(3):231–237, 1977.

G. N. Newsam and I. G. Enting.

Inverse problems in atmospheric constituent studies: I. Determination ofsurface sources under a diffusive transport approximation.

Inv. Prob., 4:1037–1054, 1988.

U.S. Environmental Protection Agency.

Guideline on Air Quality Models, Appendix W to Part 51, Title 40:Protection of the Environment, Code of Federal Regulations, 2010.

Source: http://ecfr.gpoaccess.gov.


http://ecfr.gpoaccess.gov

Documents

Mathematics of Air: Modelling of Pollutant Transport in ...humboldtcanada.com/presentations_air/stockie.pdf · Most industry-standard software is based on Gaussian plume solutions