Pollution in the Great Lakes
Hannah Conroy Broderick Charlene Gildea
Róisı́n Hill Damian O’Connor
2nd Annual Stokes Modelling Workshop
June 18, 2015
(Stokes Workshop)
Great Lakes
June 18, 2015
1 / 22
Problem description
Description of the problem
The great lakes are the main water supply for 30 million people.
Pollutants in the lakes are the main killer of marine life, encourage
the growth of algae and cause obnoxious smells.
The only feasible solution is to rely on the natural clean-up of the
lakes.
The task set was to create a mathematical model of the clean-up
process to aid policy decisions.
(Stokes Workshop)
Great Lakes
June 18, 2015
2 / 22
Problem description
Basis for single lake model
Calculated for one lake
Our assumptions
I
I
I
Constant volume in the lake
Uniform distribution of pollutant
Inflow of pollutant is constant over time
(Stokes Workshop)
Great Lakes
June 18, 2015
3 / 22
Great Lakes
Single lake model calculations
d(PV )
= Ri Pi − Ro Po
dt
P = P(t) - density of pollutant lake at time t
Ri /Ro - rate of inflow/outflow
Pi /Po - density of pollutant in inflow/outflow
V - volume
Po = P; Ri = Ro
V
(Stokes Workshop)
dV
dP
+P
= Ri Pi − Ro P
dt
dt
dV
= Ri − Ro
dt
dP
V
= Ri (Pi − P)
dt
Great Lakes
June 18, 2015
4 / 22
Great Lakes
Single lake model and solution
Ri (Pi − P) Ri
dP
=
;
=τ
dt
V
V
τ -
Volume , retention time
Rate of flow
dP
P
Pi
+ =
dt
τ
τ
Solution
P(t) = Pi + (Ps − Pi )e
−t
τ
Ps - initial density of pollutant in lake
(Stokes Workshop)
Great Lakes
June 18, 2015
5 / 22
Great Lakes
Graph of First Model P(t)/Ps
1
P/Ps
0.8
0.6
0.4
0
50
100
t (years)
150
Pollution of Lake Superior
(Stokes Workshop)
Great Lakes
June 18, 2015
6 / 22
Great Lakes
Variation for single lake model
Calculated for one lake
Our assumptions
I
I
I
Constant volume in the lake
Uniform distribution of pollutant
Inflow of pollutant is reducing
(Stokes Workshop)
Great Lakes
June 18, 2015
7 / 22
Great Lakes
Variation of single lake model
Model
dP
P
Ps e −at
+ =
; where Pi = Ps e −at
dt
τ
τ
Solution
−t
Ps (aτ e τ − e −at )
P(t) =
aτ − 1
P(t) - density of pollutant lake at time t
Ps - initial density of pollutant in lake
Volume
τ Rate of flow
ln( PPs )
n
aplanned years to achieve reduction
(Stokes Workshop)
Great Lakes
June 18, 2015
8 / 22
Great Lakes
Graph of variation of first model
1
1
0.9
P/Ps
P/Ps
0.8
0.7
0.6
0.5
0.3
0
50
100
t (years)
150
0.4
Lake Superior
(Stokes Workshop)
0
5
10
15
20
t (years)
25
30
Lake Michigan/Huron
Great Lakes
June 18, 2015
9 / 22
Great Lakes
Graph of variation of first model
1
1
0.9
P/Ps
P/Ps
0.95
0.8
0.9
0.7
0.85
0
0.5
1
1.5
t (years)
2
2.5
Lake Erie
(Stokes Workshop)
0
2
4
t (years)
6
Lake Ontario
Great Lakes
June 18, 2015
10 / 22
Great Lakes
Initial exponential increase of pollutant
Calculated for one lake
Our assumptions
I
I
I
I
Constant volume in the lake
Uniform distribution of pollutant
Initial increase in pollutant
At a certain time pollutant level decreases
(Stokes Workshop)
Great Lakes
June 18, 2015
11 / 22
Great Lakes
Initial exponential increase in pollutant
0.8
0.8
0.6
0.6
P/Ps
1
P/Ps
1
0.4
0.4
0.2
0.2
0
0
50
100
t (years)
0
150
Constant input
(Stokes Workshop)
0
50
100
t (years)
150
Decreasing input
Great Lakes
June 18, 2015
12 / 22
Great Lakes
Basis for final models
Calculated for two to four lakes, (Michigan and Huron as treated as
one lake as they both are the same height above sea level)
Our assumptions
I
I
I
I
Constant volume in each lake
Uniform distribution of pollutant
Inflow of external pollutant is constant
the direction of flow between the lakes is consistant with their height
above sea level ie: Superior to Michigan/Huron to Erie to Ontario and
finally into the Atlantic
(Stokes Workshop)
Great Lakes
June 18, 2015
13 / 22
Great Lakes
Final model
Final Model
dP1 P1
Pi1
+
=
dt
τ1
τ1
dP2 P2
V1 (P1 − Pi2 ) Pi2
+
=
+
dt
τ2
V2
τ1
τ2
P1 - density of pollutant in Lake Superior
P2 - density of pollutant in Lake Michigan/Huron
Pi1 - density of pollutant in inflow to Lake Superior
τ - Volume
Rate of flow
(Stokes Workshop)
Great Lakes
June 18, 2015
14 / 22
Great Lakes
1
1
0.8
0.8
P (density of pollutant)
P (density of pollutant)
Graph of Superior and Michigan/Huron
0.6
0.4
0.2
0
0
200
400
600
t (years)
800
0.6
0.4
0.2
1000
Input into Superior = 0
(Stokes Workshop)
0
0
200
400
600
t (years)
800
1000
Input into Superior = 0.6
Great Lakes
June 18, 2015
15 / 22
Great Lakes
Graph of Two Lakes; Superior and Michigan/Huron
P (density of pollutant)
1
0.8
0.6
0.4
0.2
0
0
200
400
600
t (years)
800
1000
Varying Superior’s input and constant input into Michigan/Huron
(Stokes Workshop)
Great Lakes
June 18, 2015
16 / 22
Great Lakes
Graph of Three Lakes; Superior, Michigan/Huron and Erie
P (density of pollutant)
1
0.8
0.6
0.4
0.2
0
0
100
200
300
400
t (years)
500
600
700
No input of pollutant
(Stokes Workshop)
Great Lakes
June 18, 2015
17 / 22
Great Lakes
Graph of Three Lakes; Superior, Michigan/Huron and Erie
P (density of pollutant)
1
0.8
0.6
0.4
0.2
0
0
200
400
600
t (years)
800
1000
Varying Superior’s input, no input to others
(Stokes Workshop)
Great Lakes
June 18, 2015
18 / 22
Great Lakes
1
1
0.8
0.8
0.6
0.6
P
P (density of pollutant)
Graph of Three Lakes; Superior, Michigan/Huron and Erie
0.4
0.4
0.2
0.2
0
0
200
400
t (years)
600
800
Varying Michigan/Huron only
(Stokes Workshop)
0
0
200
400
t (years)
600
800
Varying Erie only
Great Lakes
June 18, 2015
19 / 22
Great Lakes
Graph of Four Lakes
P (density of pollutant)
1
0.8
0.6
0.4
0.2
0
0
200
400
t (years)
600
800
No input of pollutant
(Stokes Workshop)
Great Lakes
June 18, 2015
20 / 22
Great Lakes
Conclusion
The pollutant level tends to the input pollutant level when the input
level is constant.
Rate of outflow is constant so the levels of pollutants in the lakes will
ultimately be determined by the level of input pollutants.
(Stokes Workshop)
Great Lakes
June 18, 2015
21 / 22
Great Lakes
Further areas to investigate
Varying constant inputs for multiple lake model
Exponential decrease of pollution input for multiple lake model
Diffusion of pollutants in the lakes
The effect of a sudden spike in pollutants
(Stokes Workshop)
Great Lakes
June 18, 2015
22 / 22