Ch 2.6 Case Study: Lake Burley Griffin

Lake Burley Griffin

title

Background

Lake Burley Griffin is located in Canberra, Australia's capital.

title

Background

Lake Burley Griffin was created artificially in 1962

title

Background

Serves recreational and aesthetic purposes.

title

Background

Queanbeyan is in the state of New South Wales (NSW)
Lake Burley Griffin is in the Australian Capital Territory
Safety levels/standards are different between territories.

title

Background

Extensive measurements of pollution levels (1970s)
Upstream sewage plants (Queanbeyan) contribute pollution

title

Background

Significant also was rural and urban runoff during summer rainstorms.
These contributed to dramatic increases in pollution levels
Responsible for lifting concentration levels above safety limits.

title

Background

In 1974 the mean concentration of the bacteria faecal coliform count was approximately \( 10^7 \) bacteria per \( \mathrm{m}^3 \) at the point where the river feeds into the lake.

title

Background

Safety threshold for faecal coliform count in water for contact recreational sports:

No more than 10% of total samples over a 30-day period should exceed \( 4 \times 10^6 \) bacteria per \( \mathrm{m}^3 \).

title

Use Model to Investigate

If sewage management were improved, would lake would flush out?
Would pollution levels would drop below the safety threshold?

title

General compartmental model

The lake is the only compartment.
Applying the balance law there is an input of polluted water from the river(s) flowing into the lake, or due to a pollution dump into the lake, and an output as water flows from the lake carrying some pollution with it.

title

Assumptions, Model & Solution

Lake is continuously well mixed
Pollution is uniform throughout
Lake has a constant volume \( V = 28 \times 10^6 \mathrm{m}^3 \).
Lake has constant flow F, with \( F_{in} = F_{out} \).
IVP and solution:

\[ \begin{align*} \frac{dC}{dt} & = \frac{F}{V}c_{in} - \frac{F}{V}C(t) , \,\, C(0)= c_0 \\ \\ C(t) & = c_{in} - (c_{in} - c_0) e^{-Ft/V } \end{align*} \]

Lake Pollution Model

How long will it take for the lake's pollution level to reach a specified level \( C_L \) if only fresh water flows into the lake?
Only fresh water coming in: \( c_{in} = 0 \)

\[ \begin{align*} C(t) & = c_{in} - (c_{in} - c_0) e^{-Ft/V } \\ & = c_0 e^{-Ft/V } \end{align*} \]

Solve for \( t \), using \( C = C_L \):

\[ t = - \frac{V}{F} \ln \left( \frac{C_L}{c_0} \right) \]

Lake Pollution Model: Predictions

title

Fresh water entering the lake: \( c_{in} = 0 \)
Volume: \( V = 28 \times 10^6 \mathrm{m}^3 \).
Mean monthly flow: \( F = 4 \times 10^6 \mathrm{m}^3/ \mathrm{month} \)
Initial faecal coliform count: \( c_0 = 10^7\mathrm{bacteria}/ \mathrm{m}^3 \)
\( C_L = ? \)

Lake Pollution Model: Predictions

title

Fresh water entering the lake: \( c_{in} = 0 \)
Volume: \( V = 28 \times 10^6 \mathrm{m}^3 \).
Mean monthly flow: \( F = 4 \times 10^6 \mathrm{m}^3/ \mathrm{month} \)
Initial faecal coliform count: \( c_0 = 10^7\mathrm{bacteria}/ \mathrm{m}^3 \)
\( C_L = 4 \times 10^6 \mathrm{bacteria}/ \mathrm{m}^3 \)

Lake Pollution Model: Predictions

Parameter Values
\( V = 28 \times 10^6 \mathrm{m}^3 \)
\( F = 4 \times 10^6 \mathrm{m}^3/ \mathrm{month} \)
\( C_L = 4 \times 10^6\mathrm{bacteria}/ \mathrm{m}^3 \)
\( c_0 = 10^7\mathrm{bacteria}/ \mathrm{m}^3 \)
Formula \[ t = - \frac{V}{F} \ln \left( \frac{C_L}{c_0} \right) \]

Use R for calculations:

V <- 28*10^6 
F <- 4*10^6 
CL <- 4*10^6
c0 <- 10^7
t <- - V/F*log(CL/c0)

[1] 6.414035

Lake Pollution Model

However, pure water entering the lake is not a very realistic scenario with three sewage plants and much farmland upstream.
Thus including the entrance of polluted river water into the lake model is essential.

\[ C(t) = c_{in} - (c_{in} - c_0) e^{-Ft/V } \]

title

Numerical Solutions: Maple, MATLAB, R

Including the entrance of polluted river water into the lake model is essential.

\[ \frac{dC}{dt} = \frac{F}{V} \left[ c_{in} - C(t) \right] , \,\, C(0)= c_0 \]

Case 1: \[ \begin{align*} c_{in} & = 3 \times 10^6 \mathrm{bacteria}/ \mathrm{m}^3 \\ F & = (4*12) \times 10^6 \mathrm{m}^3/ \mathrm{year} \end{align*} \]
Case 2: \[ \begin{align*} c_{in} & = 10^6 \left[ 10 + 10 \cos(2 \pi t) \right] \mathrm{bacteria}/ \mathrm{m}^3 \\ F(t) & = 10^6 \left[ 10 + 6 \sin(2 \pi t) \right] \mathrm{m}^3/ \mathrm{year} \end{align*} \]

Case 1: Maple (Ch2.5 Listing 2.7, Part 1)

title

Case 1: Maple (Ch2.5 Listing 2.7, Part 2)

title

Case 1: Maple (Ch2.5 Listing 2.7)

\[ \lim_{t \rightarrow \infty} C(t) = \lim_{t \rightarrow \infty} \left( c_{in} - (c_{in} - c_0) e^{-Ft/V }\right) = c_{in} \]

title

Case 1: MATLAB (Ch2.5 Listing 2.8)

title

\[ \begin{align*} c_{in} & = 3 \times 10^6 \mathrm{bacteria}/ \mathrm{m}^3 \\ F & = (4*12) \times 10^6 \mathrm{m}^3/ \mathrm{year} \end{align*} \]

Case 1: MATLAB (Ch2.5 Listing 2.8)

ch25Lake

title

Case 1: R (Fourth-Order Runge-Kutta)

rk4plot <- function (f, x0 , y0 , h, n) {
  x <- x0
  y <- y0
  for(i in 1:n) {
    s1 <- h * f(x0 , y0)
    s2 <- h * f(x0 + h / 2, y0 + s1 / 2)
    s3 <- h * f(x0 + h / 2, y0 + s2 / 2)
    s4 <- h * f(x0 + h, y0 + s3)
    y0 <- y0 + s1 / 6 + s2 / 3 + s3 / 3 + s4 / 6
    x0 <- x0 + h
    x <- c(x, x0)
    y <- c(y, y0)
  }
  return(plot(x,y,type="l", col="blue"))
}

Case 1: R (Fourth-Order Runge-Kutta)

f <- function(x, C) { 48/28*(3 - C) }
rk4plot(f, 0, 10, 0.1, 40)

plot of chunk unnamed-chunk-4

Case 1: R (Fourth-Order Runge-Kutta)

rk4data <- function (f, x0 , y0 , h, n) {
  x <- x0
  y <- y0
  for(i in 1:n) {
    s1 <- h * f(x0 , y0)
    s2 <- h * f(x0 + h / 2, y0 + s1 / 2)
    s3 <- h * f(x0 + h / 2, y0 + s2 / 2)
    s4 <- h * f(x0 + h, y0 + s3)
    y0 <- y0 + s1 / 6 + s2 / 3 + s3 / 3 + s4 / 6
    x0 <- x0 + h
    x <- c(x, x0)
    y <- c(y, y0)
  }
  return(data.frame (x = x, C = y))
}

Case 1: R (Fourth-Order Runge-Kutta)

f <- function(x, C) { 48/28*(3 - C) }
rk4data(f, 0, 10, 0.2, 10)

     x         C
1  0.0 10.000000
2  0.2  7.968438
3  0.4  6.526483
4  0.6  5.503016
5  0.8  4.776583
6  1.0  4.260978
7  1.2  3.895013
8  1.4  3.635260
9  1.6  3.450893
10 1.8  3.320033
11 2.0  3.227152

Case 2: Maple, Part 1 (Ch2.6 DrG)

title

Case 2: Maple, Part 2 (Ch2.6 DrG)

title

Case 2: MATLAB (Ch2.6 DrG)

title

Case 2: MATLAB (Ch2.6 DrG)

ch26Lake2

title

Case 2: R (Fourth-Order Runge-Kutta)

F1 <- function(x) {(10 + 6*sin(2*pi*x))}
c1 <- function(x) {(10 + 10*cos(2*pi*x))}
f <- function(x, C) { F1(x)/28*(c1(x) - C) }
rk4plot(f, 0, 6, 0.1, 80)

plot of chunk unnamed-chunk-7

Case 2: R (Fourth-Order Runge-Kutta)

title