Tanning leather has been conducted around the world for over 5,000 years. The leather produced was used to make sandals and boots, water containers, harnesses, bags, armour, quivers, boats, etc. The old tanning process was accomplished by using materials that were noxious and produced very unpleasant smells. Consequently, the process was usually relegated to locations that were some distance from most small villages and towns. But the process of tanning requires a significant amount of water, so tanning production had to be established near larger bodies of water (which is also where towns tend to be established).
Leather tanning is a billion dollar industry today. The process of tanning today still requires a significant amount of water. During the process, chemical biocides are added to the soaking skins to prevent bacterial growth. The polluted water used in the process is generally dumped back into the water system. With strict environmental regulations in developed countries causing greater costs to complete the tanning process, much of the actual tanning process is accomplished by shipping the hides to developing nations, where environmental regulations are much weaker and labour is much cheaper.
Your task is to create a model of a small city, dependent upon a tanning industry for many of its jobs. We will assume that the city is situated near a large river that runs into a lake a few miles away. The city was chosen because of the abundance of water needed for the tanning process. Since there is such an abundance of clean water, it is expected that this city will flourish from their new industry. Is it possible for the city to support a tanning industry and still keep water pollution to a pre-determined safe level?
Displaying the Dynamic Relationship
The diagram that will represent the dynamic relationship between sectors is a very simplified (aggregate or high-level) feedback loop diagram, connecting major sectors of a model, which captures the behaviour we expect the model to produce. At this point there are only two major concepts (sectors) to consider for this problem: the population of a city and the water pollution produced by the tanning industry that provides jobs for some of the people in the city. We will design a sector map consisting of two modules: one to represent the population of the city, and the other to represent the water pollution produced from the tanning process that is dumped into nearby rivers.
It seems reasonable to expect that as the population grows, the pollution in the water will increase. The water pollution will probably raise the death rate of the population at some point in time. We will assume that no preventative measures to control the water pollution will be enacted in advance of an indication that the pollution is causing a problem in the population. A possible sector map is shown in below.
To build this model, we will use data from another city with similar characteristics that was previously studied. For the simple population sector, there was an initial population of about 100000 people, a birth fraction of about 0.011, and a normal death fraction of about 0.01. We will use this data for our model.
For the water pollution sector, we will start with a stock labelled Water Pollution and whose inflow is labelled pollution added from tanning. We note that about 15% of the population of the city we studied was involved in the tanning industry and that each person in the tanning industry added about 0.1 units of pollution to the river each year from the work they did. We know there should also an outflow from Water Pollution entitled normal pollution absorption. We noted that about 1500 units of pollution could be absorbed in the river naturally each year. For our current model, we want to start the simulation in the past, so we can check to see if the current model will replicate some of the behaviour we saw in the city we researched. So, we will start our current model in 1970, when the tanning factory would have started production, and assume a starting average Water Pollution of 1500 pollution units.
As the tanning factory production grew (because the population grew) we can assume the pollution produced affected the death rate of the people in the city. We anticipate that the water pollution could rise as high as 10 times the initial pollution level of 1500 units. We expect the effect of pollution on the death fraction to maintain the normal death fraction when the water pollution is less than or equal to its initial level. Then it should rise fairly quickly when the ratio is greater than 1. We anticipate an S-shaped growth pattern in the death fraction multiplier up to a level 10 times the value of the normal death fraction.
Do not forget to create an actual death fraction and connect it to the deaths outflow. When Water Pollution/initial pollution is equal to 1, the death fraction multiplier should equal 1. For this model, in order to try to establish a stable population for the beginning of the model, when Water Pollution/initial pollution is equal to 2, make the death fraction multiplier 1.9.
Be sure to use the Runge-Kutta 4 integration method, with a DT of 0.125, and a simulation time of 150 years, 1970 – 2120. Make sure the non-negative check is removed from all stocks.
We would like to start this model in equilibrium, which will not be possible if the birth fraction is larger than the death fraction. So set the birth fraction to 0.01 for now and make sure that your model shows equilibrium. Look at a graph of Population (range 80000 – 110000) and Water Pollution (range 1400 – 2200) over the 150-year time span. Once your model is in equilibrium, change the birth fraction (using the STEP function) so that it will step up to its true value of 0.011 in the year 1972. This way all graphs will show equilibrium for the first two years. The STEP function has the format STEP(amount, time of change). Rerun the simulation.
- Paste of copy of your model into the box below along with a graph showing the predicted trend in Population and Water Pollution from 1970 – 2120 (make the box larger if you need to). (5 marks)
Knowing that the tanning industry would be dumping pollutants into the nearby river, the city council tried to be proactive and started building a treatment plant for the polluted water produced by the tanning process. It was ready shortly after the tanning company started operating. In our current model, rename the Water Pollution stock as Water Pollution in Treatment Plant.
The treatment plant creates a material delay between producing the pollutants and dumping the pollutants into the river. Label a new stock Pollution Remaining in Treated Water (which will have an initial value of 1500) (Good modelling practice would be to use the initial pollution level as the initial value for both Pollution Remaining in Treated Water and Water Pollution in Treatment Plant stocks).
Water will flow between the Water Pollution in Treatment Plant stock and the Pollution Remaining in Treated Water stock. We will call this flow pollutants sent to the river. The Pollution Remaining in Treated Water stock will now have the outflow that we called normal pollution absorption. You will also have to change another connection in the model to correctly incorporate the Pollution Remaining in Treated Water stock, since the Pollution Remaining in Treated Water (not Water Pollution in Treatment Plant) will now affect the death fraction.
The city council decided to specify a maximum acceptable pollution level for the river and wants to make sure the pollution in the river water does not exceed the maximum acceptable pollution level over time. Create an outflow (pointing down from the Water Pollution in Treatment Plant stock) called pollution removed. Have the simulation determine the gap between the max acceptable pollution amount (1800 pollution units) and the current Pollution Remaining in Treated Water level. Then remove the excess pollution (the amount that is over the maximum acceptable) from the Water Pollution in Treatment Plant stock. Remove the excess amount (which will be calculated as a positive value if the Pollution Remaining in
Treated Water is greater than the maximum pollution allowed) or remove zero (if the Pollution Remaining in Treated Water is less than the maximum pollution allowed, i.e., when the difference is negative). Use the MAX function with the format MAX (first amount or expression, second amount or expression). Place this MAX function within a converter that calculates the gap between Pollution Remaining in Treated Water and max acceptable pollution. We will assume that the time it takes to remove the necessary pollution is one year. This dwell time should be connected to each outflow from Water Pollution in Treatment Plant. We will assume that these pollutants are properly stored somewhere, but we will not worry about where they are stored.
Finally, make a connection from pollution removed to pollutants sent to the river, and from Water Pollution in Treatment Plant to pollutants sent to the river. We need to make sure that the pollutants sent to the river are only those pollutants left over after the treatment plant removes whatever pollution it can remove each year. The new section added to your model should look like the model section below.
- Run the simulation. Paste a copy of your model into the box below along with a graph showing the predicted trend in Population (range 80000 – 110000), Water Pollution in Treatment Plant (range 1400 – 2200) and Pollution Remaining in Treated Water (range 1400 – 2200) from 1970 – 2120 (make the box larger if you need to). Did the Pollution Remaining in Treated Water level remain at or below the maximum acceptable 1800 pollution units? (5 marks)
- Report the value for Pollution Remaining in Treated Water at the beginning of the year 2005 and in the box below and explain why pollution was above or below the acceptable level. (5 marks)
It seems as if we have captured the most important concepts for our model. But it turns out that the actual data from health records indicate that the number of deaths in the real city were different than our model simulation indicates. As a matter of fact, the city population was a little over 100100 people in 2005. So something important has been left out of the model.
After some reflection, we realize that the model assumes all information about pollution is immediately available. Realistically, data had to be collected from the polluted streams to determine how much pollution had to be removed by the treatment plant. To accomplish such a task it would have been necessary to get permission from the city council and write some grants to obtain funding to support the data collection, a process that can take 1 year. Another 0.5 years would be needed to collect the data from the various parts of the river and the lake downstream. To include the delay involved with the data collection process in the model, we will set up a first order information delay (total of 1.5 years). This delay occurred as scientists were taking information from the pollution in the river and then trying to determine the gap or difference between the maximum acceptable pollution amount and the actual pollution remaining in the treated water. We will use a converter (called pollution readings delay collecting) to represent this information delay, rather than the whole information delay structure. To include the delay we will use the SMTH1 function. The SMTH1 function is defined as: SMTH1(amount that will be delayed, delay time). First, remove the connection between the Pollution Remaining in Treated Water and the gap converter. The delay should be placed between the Pollution Remaining in Treated Water stock and the gap converter, since it is the collection of pollution readings that is being used to determine how large the gap between the water pollution in the river and the maximum acceptable pollution amount actually is. The section added to your model to simulate the information delay should look like the model section below.
- Rerun the simulation. In the box below explain how the model behaviour changed for the Pollution Remaining in Treated Water graph compared to your previous model that didn’t have the information delay. Also explain why the behaviour changed if it did. (5 marks)
Not only did it take 1.5 years to find funding and collect the water samples from the river and lake in the area around the city, it took another 0.5 year for the report to be written about the current level of pollution in the river water and for the report to work its way up the chain of command to the correct administrators, so they could decide how to deal with the extra pollution. Place another first order information delay between the gap and the pollution removed flow, since the pollution level cannot be modified until the city council has received the scientist’s report. The delay here occurred after the gap (between the Pollution Remaining in Treated Water level and the maximum acceptable pollution amount) had been calculated, but before any actual modification of pollution removal was begun. (Again, use the SMTH1 function.) Call this converter pollution to remove delay in acting.
Although we are gaining ground on including the necessary information in our model, there is another problem to address. A treatment plant has a limit on the amount of pollution it can process. To incorporate this limit, we will remove the connection between pollution to remove delay in acting and the pollution removed flow. Create a new converter called fraction of pollution to remove. The new converter depends upon the Water Pollution in Treatment Plant and on the pollution to remove delay in acting amounts. The converter will specify the fraction to remove as either 0.5% or the fraction of pollution that the scientist’s report indicates should be removed, whichever is smaller. This fraction is determined by the pollution to remove delay in acting divided by the current Water Pollution in Treatment Plant level. Use a MIN function to determine the smallest fraction to remove. The format is MIN (first amount or expression, second amount or expression). Place the MIN function in the fraction of pollution to remove converter. The pollution removed flow will now depend upon the Water Pollution in Treatment Plant, the fraction of pollution to remove converter, and the time to remove pollution converter.
- Rerun the simulation. Paste a copy of your final model into the box below along with a graph showing the predicted trend in Population (range 80000 – 110000), Water Pollution in Treatment Plant (range 1400 – 2200) and Pollution Remaining in Treated Water (range 1400 – 2200) from 1970 – 2120 (make the box larger if you need to). Also report the predicted values for Population, Water Pollution in Treatment Plant and Pollution Remaining in Treated Water for the beginning of the year 2005 (make the box larger if you need to). (5 marks)