Ch 2.5 Lake Pollution Models

One of several defining characteristics of our planet is that there is a lot of water. According to [1],

• About 71 percent of the Earth's surface is water-covered.
• The oceans hold about 96.5 percent of all Earth's water.
• Water also exists in the air as water vapor, in rivers and lakes, in icecaps and glaciers, in the ground as soil moisture and aquifers, and in the life forms that populate Earth.

Since life on Earth relies heavily on water, the safety and cleanliness of the water is a natural topic of interest for us.

In this section, we introduce two simple models for contaminant movement (advection) through a well-mixed body of water by considering two cases from our book [7]:

• Salt solution in a tank
• Lake pollution

The advection-only case leads to an ODE model and is a good place to start.

• The advection-diffusion-reaction (ADR) case leads to PDE models and are revisited for the lake pollution case in Ch12.4.

What don't ants get sick?

They have anty-bodies.

• We will follow the steps of our modeling process for this example.
  
• The numerical solution will be illustrated in the case study described in Ch2.6.

A common usage of salt water is the production of brine. According to Wikipedia [2],

• Brine is a high-concentration solution of salt in water.
• Brine may refer to the salt solutions ranging from about 3.5% (seawater) up to about 26% (saturated).
• Brine is used for food processing and cooking (pickling and brining), for de-icing of roads and other structures, and in a number of technological processes.

• For our example, we will consider a tank containing salt water.
• There is an inflow pipe where salt water enters the tank, and there is an outflow pipe were salt water leaves the tank.
• The concentration of salt in this incoming flow of salt water varies with time.
• The tank is well-mixed and salt solution flows out at same rate that it flows in (tank volume remains constant).
  

Determine the amount of salt (the solute) dissolved in a tank of water (the solution) as a function of time.

• The tank contains 100 liters of salt water.
• Salt water flows into the tank at a rate of 10 liters per minute.
• The salt solution flows in and out at same rate.
• The tank is well-mixed, so that salt in the tank is uniformly distributed with no salt clumps forming or other pockets of higher concentrations of salt.
  
  • This allows us to determine the salt levels in the tank as a function of time, rather than on time and location, using an ODE model rather than a PDE model.
    

The input-output compartmental diagram for salt solution is:

From the balance law and compartment diagram, we have

• Let \( V=100 \) denote the volume (liters) of the tank.
  
• Let \( S(t)= \) amount (kg) of salt in tank at \( t \) minutes.
• Let \( S(0)=s_0= \) initial amount (kg) of salt at time \( t=0 \).
• Let \( c_{in}(t)= \) concentration (kg/liter) of salt in the incoming flow of salt water.

The rate at which salt enters tank is the product of flow rate in and concentration of salt in the incoming mixture:

\[ \left(10 \frac{L}{min}\right) \left(c_{in}(t) \frac{kg}{L}\right) = 10 c_{in}(t) \frac{kg}{min} \]

The rate at which salt leaves tank is the product of flow rate out and concentration of salt in the outgoing mixture is given by

\[ \left( 10 \frac{L}{min}\right) \left(\frac{S(t)}{100} \frac{kg}{L} \right) = \frac{S(t)}{10} \frac{kg}{min} \]

Our IVP will then have the following form:

\[ \frac{dS}{dt}= 10 c_{in}(t) - \frac{S(t)}{10}, \,\, S(0) = s_0 \]

The Normal form of the IVP is

\[ \frac{dS}{dt} + \frac{1}{10}S(t) = 10 c_{in}(t) , \,\, S(0)= s_0 \]

• We next find the exact solution to the IVP using the method of integrating factors.
• Recall that this method can be used for a linear first order ODE of the following form, using the steps shown below:

\[ \begin{aligned} \frac{dy}{dt} + p(t)y & = g(t) \\ \mu(t) & = e^{\int p(t)dt } \\ y(t) & = \frac{\int_0^t \mu(s)g(s) ds}{\mu(t)} + \frac{C}{\mu(t)} \end{aligned} \]

Thus for the salt tank IVP, we have

\[ \begin{aligned} \frac{dS}{dt} + \frac{1}{10}S(t) &= 10 c_{in}(t) , \,\, S(0)= s_0 \\ \mu(t) & = e^{\int \frac{1}{10} dt } = e^{\frac{t}{10} } \\ S(t) & = \frac{\int_0^t 10 e^{\frac{s}{10} }c_{in}(s) ds}{e^{\frac{t}{10}}} + \frac{C}{e^{\frac{t}{10}}} \\ S(t) & = e^{-\frac{t}{10} } \int_0^t 10 e^{\frac{s}{10} }c_{in}(s) ds + s_0 e^{-\frac{t}{10} } \end{aligned} \]

Methods of integration are required to solve for \( S(t) \), the specifics of which depend on the form of \( c_{in}(t) \).

Suppose \( c_{in}(t) = c_1 \). Recalling that

\[ \small{ \int e^{Ax }dx = \frac{1}{A}e^x + C } \]

we can solve our IVP as follows:

\[ \small{ \begin{aligned} S_1(t) & = e^{-\frac{t}{10} } \int_0^t 10 e^{\frac{s}{10} }c_{in}(s) ds + s_0 e^{-\frac{t}{10}} \\ & = 10c_1 e^{-\frac{t}{10} } \int_0^t e^{\frac{s}{10} } ds + s_0 e^{-\frac{t}{10}} \\ & = 10c_1 e^{-\frac{t}{10} } \cdot 10\left( e^{\frac{t}{10} } - 1 \right) + s_0 e^{-\frac{t}{10}} \\ & = s_0 e^{-t/10} + 100c_1(1-e^{-\frac{t}{10} }) \end{aligned} } \]

For \( c_{in} \) as below, integration by parts can be used:

\[ \small{ \begin{aligned} c_{in}(t) & = 0.2 - 0.1\sin(t) \\ S_2(t) & = e^{-\frac{t}{10} } \int_0^t 10 e^{\frac{s}{10} }c_{in}(s) ds + s_0 e^{-\frac{t}{10}} \\ & = s_0 e^{-t/10} + 20 + \frac{10}{101}(\sin(t) - 10\cos(t) - 192e^{-t/10}) \end{aligned} } \]

• For a tank containing a salt solution in a controlled setting, the assumptions given at the beginning of the model are reasonable.
• The fixed volume of the tank is a known value, as is the initial amount \( s_0 \) of salt in the tank.
• The flow rate of the incoming salt solution can be precisely governed with a valve, and similarly with the incoming concentration \( c_{in}(t) \) of salt in that flow.
  
• A stirring mechanism can be used to insure that the solution in the tank is well mixed.
  

• The analytical solution derived from the assumptions is

\[ \small{ S(t) = e^{-\frac{t}{10} } \int_0^t 10 e^{\frac{s}{10} }c_{in}(s) ds + s_0 e^{-\frac{t}{10}} } \]

• The term associated with the initial condition \( s_0 \) is transient.
  
• The graphs tend towards a limiting behavior largely dependent on concentration \( c_{in}(t) \) in the incoming flow.
  

• Using the salt solution in a tank as a guide, we next develop a simple lake pollution model.
• We will follow the modeling steps as before, while exploring the numerical solution in Ch2.6.

• In Grand Junction, we rely on Grand Mesa reservoirs as a safe source for our drinking supply, although regional drought conditions have required us to draw from the Colorado River as well.
• In general, lakes and rivers are typical sources of drinking water around the world.
• However, pollution in our lakes and rivers has become a major problem, particularly over the past 50 years.
• In order to improve this situation in the future, it is necessary to gain a good understanding of the processes involved.

According to the EPA,

• Nutrient pollution builds up in our nation's lakes, ponds, and streams.
• Both algae and high nitrate levels cause problems in sources of drinking water.
• EPA's 2010 National Lakes Assessment found that almost 20 percent of the 50,000 lakes surveyed had been impacted by nitrogen and phosphorus pollution.
• The report also showed that poor lake conditions related to nitrogen or phosphorus pollution doubled the likelihood of poor ecosystem health.

• Point source pollution (PSP) includes contaminants entering a water body that can be traced back to a specific source, location, and offender.
• Examples include industrial waste, output from sewage treatment facilities, hazardous chemical deposition, and heat.
• Heat acts as a pollutant, for example when heated water (used as a coolant in power plants) is released into nearby lakes and alters the water temperatures.
• Non-point source pollution (NPSP) cannot be traced back to a specific source, location, and offender.
• NPSP comes from many diffuse sources and often enters in small amounts but can become concentrated in lakes and other freshwater resources.

• Some way of predicting how the situation might improve (or decline) as a result of current management practices is vital.
  
• For example, strategies for Oneida Lake in New York State included barnyard runoff management systems, manure storage systems, and nutrient and sediment control systems.
• These practices lead to reduced phosphorus loads to the lake, and were successful enough that the lake could support aquatic life and recreation.
• Other long-lived pollutants include DDT, PCBs, and mercury can remain for decades are their release has stopped.
• Thus it is of interest to predict how pollutant amounts or concentrations vary over time and under different management strategies.

Determine the concentration of pollutant in the lake as a function of time, and use it to determine the length of time required for mitigation effects to reduce pollution levels by a specified amount.

• The lake contains a fixed volume of water.
• Water flows into the lake at the same rate that it flows out, allowing the lake volume to remain fixed.
• The pollutant of interest enters and departs the lake with the water flow. This could be considered as an aggregrate of all ways in which pollution enters and leaves the lake.
• Concentration of pollutant in the flow entering lake is constant; for example, an average value.
• The water dynamics of the lake are such that it is well-mixed.
  

The compartment diagram is similar to the salt water tank:

Using the balance law and the compartment diagram, we have:

\[ \small{ \begin{Bmatrix} \mathrm{rate \, of \, change} \\ \mathrm{of \, concentration \, of } \\ \mathrm{pollutant \,in \, lake } \\ \mathrm{at \, time \,} t \end{Bmatrix} = \begin{Bmatrix} \mathrm{rate \, of \, change} \\ \mathrm{of \, concentration \, of } \\ \mathrm{pollutant \, entering \, lake } \\ \mathrm{at \, time \,} t \end{Bmatrix} - \begin{Bmatrix} \mathrm{rate \, of \, change} \\ \mathrm{of \, concentration \, of } \\ \mathrm{pollutant \, leaving \, lake } \\ \mathrm{at \, time \,} t \end{Bmatrix} } \]

• \( V= \) volume of the lake, in \( m^3 \).
• \( F= \) rate at which the water flows into and out of the lake, in \( m^3/day \).
• \( M(t)= \) mass of pollutant in lake at time \( t \), in \( gm \).
• \( C(t)= M(t)/V= \) concentration of pollutant in lake at time \( t \), in \( gm/m^3 \).
• \( c_0= \) initial concentration of pollution in lake.
• \( c_{in}= \) concentration of pollutant in flow entering lake, in \( gm/m^3 \).

The rate at which pollutant enters & leaves lake is product of flow rate in and concentration of pollutant in water flow:

\[ \small{ \begin{aligned} \left(F \frac{m^3}{day}\right) \left(c_{in}(t) \frac{g}{m^3}\right) &= F c_{in} \frac{g}{day} \\ \left(F \frac{m^3}{day}\right) \left(C(t) \frac{g}{m^3}\right) & = F C(t) \frac{g}{day} \end{aligned}} \]

To convert these mass expressions into concentrations, we divide by \( V \) and write our IVP as

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

The IVP is given by

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

The exact solution can be found using either by the method of integrating factors or by separating variables (see text):

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

Observe that \( C(t) \) tends to the value of the incoming concentration \( c_{in} \) as \( t \) gets large:

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

• Suppose that mitigation efforts are 100% successful, so that only fresh water flows into the lake.
• How long will it take for the lake's pollution level to reach 5% of its initial level? In this case, we use \( c_{in} = 0 \):

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

Next, let \( C(t) = 0.05c_0 \) and solve for \( t \):

• See next slide

\[ \begin{aligned} C(t) &= c_0 e^{-Ft/V } \\ 0.05c_0 &= c_0 e^{-Ft/V } \\ \ln(0.05) &= - \frac{Ft}{V} \\ t &= - \frac{V}{F}\ln(0.05) \\ t &\cong \frac{3V}{F} \end{aligned} \]

-log(0.05)

[1] 2.995732

\[ V = 4.58 \times 10^{11} m^3,\,\, \,\, F = 4.80 \times 10^8 \frac{m^3}{day}\left(\frac{365 \, day}{year} \right) \]

Thus Lake Erie requires almost 8 years to clear up to within 5% of its current levels; see R computation below:

\[ t \cong \frac{3V}{F} \, \mathrm{years} \]

-log(0.05)*(4.58*10^(11))/(4.8*10^8*365)

[1] 7.831309

\[ V = 1.636 \times 10^{12} m^3,\,\, \,\, F = 5.72 \times 10^8 \frac{m^3}{day}\left(\frac{365 \, day}{year} \right) \]

Thus Lake Ontario requires almost 24 years to clear up to within 5% of its current levels; see R computation below:

\[ t \cong \frac{3V}{F} \, \mathrm{years} \]

-log(0.05)*(1.636*10^(12))/(5.72*10^8*365)

[1] 23.47456

• The assumptions for this model have led to a simple representation of the movement of pollution through a lake by advection processes only.
• The concentration of pollutant in the lake tends towards the value of the incoming concentration over time.
  
• The predictions from these models should be taken only as preliminary, as our assumptions limit accuracy of results.
• One clear over-simplification is that lake water is well-mixed.
  
• Due to the lack of well-mixed water in reality, the time taken to flush the lake will typically be much longer due to the slower processes of diffusion (Ch9.8) and reaction.
• The flow rate \( F \) and concentration \( c_{in} \) will also likely have seasonal variation, rather than being constant.

[1] How Much Water is There on Earth?, https://www.usgs.gov/special-topics/water-science-school/science/how-much-water-there-earth, retrieved on 9/2/2022.

[2] Brine, https://en.wikipedia.org/wiki/Brine, retrieved on 9/2/2022.

[3] Water Supply, https://gjcity.org/297/Water-Supply, retrieved on 9/2/2022.

[4] Drought Forces Grand Junction To Dip Into Colorado River, https://www.cbsnews.com/colorado/news/drought-grand-junction-colorado-river/, retrieved on 9/2/2022.

[5] Where This Occurs: Lakes and Rivers, https://www.epa.gov/nutrientpollution/where-occurs-lakes-and-rivers, retrieved on 9/2/2022.

[6] Pollution, https://www.lakescientist.com/pollution/, Sarah Hicks, Kent State University, retrieved on 9/2/2022.

[7] Mathematical Modeling with Case Studies, Barnes and Fulford, CRC Press, 2015.