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Introduction
This series of investigations introduces students to the principles
of system dynamics, and to computer software (Stella II) that
enables students to model the behavior of natural and human systems.
System dynamics is a collection of ideas and thinking skills that
helps learners to consider the world as a set of interconnected
parts. It focuses on the ways in which different components of
systems -- for example, populations of grasses, rabbits and lynxes
in an ecosystem -- interact with one another to generate the system's
behavior as a whole. Concepts such as stocks, flows and feedback
loops help to explain the sometimes counterintuitive, complicated
behavior of many systems across nature. System dynamics therefore
promotes a set of "generic" thinking skills, one that
can be applied to a wide range of phenomena within and across
many disciplines.
The principles of system dynamics really come alive, however,
when students can actually apply and explore them through the
investigation of real phenomena. Here, Stella II software is particularly
valuable as a vehicle for enabling students to build, express,
and test their understanding of systems within a user-friendly
but rigorous environment. Stella allows students at many levels
to build models of systems -- such as predators and prey in an
ecosystem -- both by making iconographic representations of the
real system they are modeling, and by specifying the functional
mathematical relations that bind together the components of the
model/system.
This set of investigations seeks to introduce students to the
concepts of system dynamics, the Stella II software environment,
and the application of the software and concepts to one cross-disciplinary
issue: namely, whether Tamarack Pond in the Black Rock Forest
can support the water demands of the town of Cornwall, NY (or
an equivalent watershed-township in another area of the country).
This investigation is therefore intended to cultivate a deeper
sense of how to build and test integrated understandings of the
natural and human world, drawing on new thinking skills, software
tools and the more familiar data from the Black Rock Forest available
throught the Data Harvester.
Insights/Curriculum Highlights:
A system is a set of components that interact with one another, leading to changes over time in the system's behavior.
The only "complete" system is the universe itself. Smaller "subsystems" are defined by questions and problems that are of interest or consequence to human beings. These questions delineate what components and interactions are relevant to understand and pay attention to.
The structure of a system -- i.e., how its components relate to one another -- determines how it behaves.
Systems in nature share similar structures, and thus similar patterns of behavior. Therefore, techniques to characterize and understand generally how systems behave can be applied to the study and analysis of phenomena/problems that span across many disciplines.
A model is a symbolic representation of the components and interactions that make up a system. Mathematical equations can be models; so can diagrams, paintings and poems. There are many different types of models that people use to represent their understanding of a system; some techniques are better than others, depending on the goal of the modeling exercise.
Dynamic simulation models are computer-based representations of natural or human systems. Components and interactions are specified both by visual representations (for example, as stocks, flows and parameters) and by mathematical equations. The computer can solve all of the equations that describe a system very quickly; simulation software therefore enables a student to explore how systems behave and change over time.
A modeling exercise should always begin with a clear question or problem that the modeler wishes to address: The question, when stated clearly and precisely, determines where the boundaries of the system lie in space and time, and what components and interactions really matter.
Models can be built in many ways. One effective method is a combination of direct observation of nature, hands-on laboratory experimentation, and dynamic simulation modeling.
Whether a pond-stream system is able to support the water needs of a small town depends on many interacting factors, including: the amount of rainfall into the pond and its watershed; the size of the pond and watershed; the area of the drainage stream leading out of the pond; the per capita water use of the town; the population of the town; and any policies the town might institute to change its consumption patterns as the water supply changes.
All of these factors can be represented in words, diagrams and
mathematical expressions within a simulation model. A simulation
model of the Tamarack pond-stream system reveals many complicated
behaviors that would not necessarily be apparent at first glance.
Thinking Skills/Pedagogical Highlights:
Learning formal systems thinking concepts and basic software operations in Stella II environment.
Making connections between real world observations of systems, lab experiments that illustrate the basic scientific concepts operating in those systems, and dynamic models that integrate those concepts dynamically.
Learning and applying basic physics concepts, through hands-on investigation, into a modeling environment.
Building a quantitative model of a natural system using real data.
Building a quantitative model of a human system by making educated assumptions based on the students' own experience and assumptions.
Integrating the natural system and human system model into one complete pond-stream-town system model.
Assessing the behavior of a model in light of the question motivating the modeling exercise, and with reference to the model structure.
Thinking about many interacting parameters simultaneously.
1. Introductory exercise: What is a system? What is a model?
Students learn some of the basic, central concepts of systems thinking/system dynamics. Papers, books and web sites that are particularly good for this include:
- http://sysdyn.mit.edu
- http://www.teleport.com/~sguthrie/cc-stadus.html
2. Introductory exercise: Modeling in Stella II (the bathtub)
There are also a number of good introductory materials for getting started with modeling in Stella II, available at the above WWW sites.
Additionally, teachers can lead students through the building
of a simple bathtub model, as during the summer workshop.
3. Modeling Tamarack Pond 1: What do we need to know to build a model?
The first step in building a good model is to start with a good question. In our case, we ask: Can Tamarack Pond support the water needs of the town of Cornwall for a one to two-year period of time? For schools in different areas of the country, this investigation can be adapted with small changes to other types of watersheds (including arid regions where groundwater, not reservoirs, supplies water for many people). The teacher can either present the question directly, or introduce it to students with an initial discussion.
Once the purpose of the modeling investigation is clear, an important first step is to think about the physical system itself, and what aspects of it are essential to understand in order to build a model that will help address the question that motivates the investigation. One way to begin this is to ask students to draw (or lead the teacher through drawing on the board) a picture of the pond-human system. (For students in arid sections of the country, such as Marana, AZ, the system in this case would be the groundwater lying underneath Tucson.) After completing the picture, the teacher can pose the following questions to the class, an exercise aimed at having students identify what must go in to the final system model, and what kind of system model "building block" each component and interaction represents.
Questions to ask:
- What are the reservoirs (stocks) in the system to be modeled?
- What are the flows? Where do they come from and where do they go to (i.e., from outside the system, among and within the system's stocks, or flowing outside the system)? These questions are important, as they force students to define the boundaries of the system.
- What do the flow rates depend on? For example, in the flow of water into the pond, is it just the area of the pond that determines how much rainfall flows in, or must one take into account the area of the watershed as well?
- Which relationships in the system (i.e., between stocks and flows and flows and parameters) do we know how to describe using mathematical descriptions? Which ones do we not? What external data would our model need in order to run?
* Here, students may not feel confident that they can express the interactions in the system in terms of mathematical descriptions. Some attention should be paid in working back and forth between the pictoral representation of the pond on the blackboard, verbal descriptions of how the pond-stream components interact with one another, and logical mathematical statements that describe those interactions quantitatively. A good example of this is the flow of water into the pond: Verbally, students will be able to reason that the total amount (i.e., volume flow) of water flowing into Tamarack is a function of the amount of rainfall that falls in a given period and the combined area of the pond and its watershed. A helpful way to turn this into a "quantitative" statement is to go through the units of measure that are associated with each part of this sentence. In this case, for example, we wish to find an expression for the total flow (i.e., volume) into the pond in a given period of time (say in a month). This implies that the units of measure are in some unit of volume (e.g., meters cubed, or m^{3}) per unit time (months). So, how would you relate the rainfall (expressed in meters / month) and the area of the pond and watershed (meters squared, or m^{2}) so that the final units were in m^{3}/month? This kind of exercise can and should be repeated as the class defines each of the functional relations between the model's components.
- What simplifying assumptions are we making here, and how do we justify those assumptions?
- Through these questions, the teacher should keep in the back of her/his mind the final model that was developed of Tamarack during the summer workshop. The teacher can subtly guide the group down more productive lines of reasoning, while at the same time fostering creativity and improvisation within the class.
A good thing to do while going through these questions is to write down the students' answers on a blackboard or whiteboard, discussing and debating as you go. At the end of this portion of the investigation, the class should have a "master list" identified of all the stocks and flows in the system and how the rates of the flows change -- both with respect to variables external to the system (such as rainfall) and those intrinsic to the system (such as the changing depth of water in the pond). Many of these relations can be expressed quantitatively, though there will be at least one important relationship that will not (see below); additionally, students will need to identify what external data will be required to run the model eventually.
Once the class gains a sense of what the model components are
and how they interact with one another, a problem should emerge:
How can we express changes in the flow of water out of the pond
(i.e., in the drainage stream) as a function of the lake level,
which itself changes with rainfall? Working this problem out is
the subject of the next investigation.
4. Hands-on and modeling investigation: The Torricelli bucket
In this investigation, students undertake a laboratory investigation of the "Torricelli bucket" to understand the relationship between pond depth and drainage stream flow. Students will emerge with both a qualitative and quantitative understanding of how the flow of water out of a bucket depends on the depth of the water above the drainage.
In the lab, take a bucket with a drainage outlet at its base and mark off lines of depth at fixed intervals with a marker (every 5 cm, for example; see Figure 1). Then fill the bucket with water to the top depth marker (keeping the opening at the base plugged up). Our goal will be to investigate how changes in the water level above the base influence the rate of flow out of the bucket.
The bucket can be seen as a simplified abstraction of the pond-stream system, the bucket representing the pond, and the drainage the stream. By looking at the bucket-drainage system in this way, students can isolate and investigate the basic physical relationships between pond depth and stream flow at work in the laboratory.
The experiment consists of students opening up the drainage and time how long it takes for the bucket to drain from the top depth to the next level. Mark the time in a lab notebook or equivalent. Then repeat for the next increment of depth, and the next, and so on. Questions/points to note:
- What happens to the time it takes for the bucket to drain an equal increment of depth as the total depth of the bucket decreases?
- Make a graph of the flow out as a function of depth in the bucket. What kind of relationship does this look like?
- What would happen to the flow rate if you increased or decreased the size of the opening at the base of the bucket? Make some guesses about how much faster or slower the water would flow with a small increase or decrease in area and mark them down.
After completing this, discuss with students how the depth of the water in the bucket affects the flow rate. Why do you observe the decrease in flow rate with decreasing depth? What forces are acting on the water? What else affects the rate of flow out of the bucket (Hint: remember that "flow," as we have been using the term, refers to a volume of water flowing -- a quantity expressed volume/unit of time)?
After students have gained a sufficient qualitative understanding of the factors involved in controlling the rate of flow out of the bucket, introduce Torricelli's equation:
* Flow (in m^{3}/second) = Area of opening * (2*g*depth of water (in m))
where g = acceleration due to gravity (=9.8 m/s^{2}).
Now armed with both a qualitative and a quantitative understanding of how water depth controls the flow out of the Torricelli bucket, students can build a Stella model that simulates the bucket's behavior. Students should first build a qualitative model (i.e., no equations or data, just the model structure) of the bucket-drainage system.
- What is the reservoir? In what units is the reservoir expresses?
- What is the flow? What are its units?
- What are the parameters and their units?
- What should the time specifications of the model be?
- One example of a way in which the Torricelli bucket model could be constructed is given in Figure 2.
Then, for each model interaction that can be described by an equation, students should enter the appropriate mathematical expression. What additional data are needed? Students should decide what measurements must be made to the bucket in order to have the relevant data for their model. Once obtained, students can go ahead and run their model.
View a graph of flow rate as a function of depth. To do this
deploy a graph pad onto the model and double click on it. In the
dialogue box that pops up (Figure 3), choose to view a scatterplot,
with the flow rate on the Y-axis and the bucket depth on the X-axis.
How do the model results compare with the experimental results?
What are the possible sources of error?
5. Modeling Tamarack Pond 2: Torricelli and Tamarack
Not completely unwittingly, students have now built perhaps the most conceptually challenging portion of the complete Tamarack Pond model. In this portion of the investigation, students will turn their Torricelli bucket model into a model of the Tamarack Pond-stream system it represented in simplified form.
Now we wish to investigate the dynamics of the pond-stream system at a monthly time scale; hence, in the "Time Specs" dialogue box the units should be changed to "Months," and the "Length of Simulation" changed from 0-12 (corresponding to 12 months of rainfall data from the Black Rock Forest Ridgetop sensor).
Students should also rename the bucket model components so that they now represent the actual Tamarack Pond system. The names and some of the numbers may change (see Table 1 below for useful data and parameters), but the model structure does not.
Now the model captures how water flows out of Tamarack; we have still not expressed how water flows in into the pond from outside the system. Working from the picture on the blackboard and the list of model components and interactions developed earlier, students should add the appropriate model components, and their corresponding equations and initial values, to their Tamarack Pond model diagram. Refer to Table 1 for useful data. One example of what the diagram could look like is given in Figure 4.
One final gap in the model is the absence of rainfall data. Students should obtain monthly rainfall data for 12 months from the Black Rock Forest Ridgetop sensor. To do this, go to the Data Harvester and choose to see all of the Ridgetop sensor precipitation data as a time series (Figure 5). Select to view a date range with continuous data coverage (most likely between April, 1996-June, 1997), then select "Data Table." This will automatically download the data you are viewing into an Excel spreadsheet. Once the data are in the spreadsheet, students should tally up monthly totals for a 13-month period of interest (remember, the simulation will run from 0-12 months, so we actually need a total of 13 months worth of data).
Now we must get the 13 months of rainfall data into the model itself. Double-click on the parameter that was created for rainfall in the model. From the "Builtins" menu on the right choose "Time." Then click on "Become Graph." As shown in Figure 6, the Input column refers to the month in the simulation (from 0-12). For each month, place the cursor into the Output column and enter the data value for that month (the student needs to remember what the starting month is).
Now the model should be ready to run. Deploy graph pads on the model to see how different parameters vary through time. Some questions:
- Do you expect that the stream flow rate will vary with pond depth in the same way as your Torricelli bucket? View a scatterplot of stream flow vs. pond depth. Were your expectations confirmed? Why or why not?
- Select to view rainfall, pond depth, and stream flow on the same graph. What patterns do you notice. When rainfall goes to its lowest value, how long after does it take for the pond depth to get to its minimum? What about stream flow? Why is this?
- What would happen if you were to increase the cross-sectional area of the drainage stream? What if you decreased the area of the stream? Test your guesses using the model.
- What important processes are left out of the model, and how would their inclusion affect the model's behavior?
- Try getting more precipitation data from the Ridgetop and running the model for a longer period. Just on the basis of the additional rainfall data, what do you expect to happen? (Remember to change the Length of Simulation before you add more data.)
- On ths basis of your model's behavior so far, do you believe
that the volume of water flowing out of Tamarack will be enough
to support the water needs of the town?
Quantity | Size | Unit |
Tamarack Pond area | 73,000 | m^{2} |
Tamarack Pond watershed area | 380,000 | m^{2} |
Acceleration due to gravity (g) | 2.55 x 10^{7} | m / month^{2} |
Initial volume of water in Tamarack | 1.46 x 10^{5} | m^{3} |
Cross-sectional area of stream | .0042 | m^{2} |
Table 1: Tamarack Pond data, and some other useful parameters
6. Modeling Tamarack Pond 3: What about the town?
In the previous investigations students have built a working model of the pond-stream system of Tamarack, and have used real data from the Black Rock Forest to investigate the dynamic behavior of that system over the past couple of years. In this investigation, we will now turn our attention to the town itself, with the goal of estimating and modeling how much water the town consumes every month.
First, ask students to estimate their monthly water consumption. This can be done as a himework assignment, where the student monitors her/his water use over the course of a day and uses this information to make an "order-of-magnitude" guess. In class, compare the students' estimates.
Then make an educated guess about the likely size of a small town like Cornwall, NY.
Then ask students to express, first in words and then in a mathematical expression, what the total monthly water consumption (in m^{3} / month) of the town is from their per capita estimates (in m^{3} / person / month) and their estimates of the town population (in # of people). Using this information, have students build a model of the town's monthly water consumption, where the town consumes water by draining a central reservoir (see Figure 7 for an example) that has an initial volume of 300,000 m^{3} of water.
Run the model. Under the current set of assumptions about per
capita water use and the town's population, how long does it take
for the town to drain its water supply?
7. Modeling Tamarack Pond 4: Putting it all together!
With the model of Part 6 complete, it becomes clear that the consumtion of water by people has to be offset by the production of drinkable water by nature -- i.e., some flow in to the reservoir that is at least equal to the flow out from human use. If the rate of human consumption exceeds the rate of natural (re)generation, then sooner or later the existing stock of the resource will be consumed. This is true for all natural resources.
So, in our case, we want the original source of water for the town to come from the Tamarack drainage stream. Students should combine the Tamarack model with their human water consumption model, so that the stream flow out of the pond becomes the flow in to the town reservoir.
Now run the pond-stream-town water consumption model. Questions and additional experiments:
- Does the town run out of water during the simulation? If so, how long does it take?
- For how long does the stream flow (i.e., the flow into the town reservoir) equal the town water consumption (i.e., the flow out)? If the town were getting all the water it needed from the stream, the stream flow divided by the town water consumption would equal 1. Find a way for your model to make this calculation. For how long is the fraction 1?
- What would happen if the town were to institute conservation
measures in the event that the reservoir got too low? For example,
what if the town began to decrease its per capita consumption
incrementally after the reservoir dropped below some critical
level. How could you build such measures into your model? Try
it!!!
Figures
Figure 1: The Torricelli Bucket
Figure 2: Example of possible Torricelli bucket model structure
Figure 3: Screen shot of Stella dialogue box for choosing a scatterplot
of Torricelli model output.
Figure 4: Example of possible Torricelli-Tamarack model structure
Figure 5: Time-series precipitation data from Ridgetop sensor
Figure 6: Screen shot of Stella dialogue box for entering monthly
rainfall values from Ridgetop sensor
Figure 7: Possible model structure for town water consumption