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\title{\ifnum\TeacherEdition=1 TEACHER VERSION  \else STUDENT VERSION \fi\\ Don't Drink the Kool-Aid\textsuperscript{\textregistered}!}





\markboth{ Don't Drink the Kool-Aid\textsuperscript{\textregistered}!}{ Don't Drink the Kool-Aid\textsuperscript{\textregistered}!}


\author{
Kristin Burney\teacher{$^1$}\!\!\! , Lydia Kennedy\teacher{$^2$}\!\!\! , Audrey Malagon\teacher{$^3$}\\
Virginia Wesleyan College\\
Norfolk VA  23502 USA\\
\teacher{(1) kburney@vwc.edu, (2) lkennedy@vwc.edu, (3) amalagon@vwc.edu}
}



\makeStitlePDFLaTex


\begin{abstract}
Single-compartment mixing is an important foundational component of any study of ordinary differential equations. Typically, problems utilize salt as the solute. In this modeling scenario, use of colored drink powder as the solute enables students to observe a color change as the mixing progresses. Students run two experiments that involve a concentrated solution of drink powder flowing from an upper tank to a lower tank, initially filled with plain water. They make qualitative observations about the intensity of the color in the tanks and record initial conditions. In the first experiment, the spout on the lower tank remains closed; in the second, the solution flows from both tanks at the same flow rate. Prior to experimentation, students are asked to predict the outcomes and sketch results. Following experimentation, students develop mathematical models and use them to determine the amount of drink powder in the lower tank at the end of each experiment. Specifically, first-order separable and linear differential equations are employed. Additional components of the lab invite students to consider a situation in which tank outflow rates differ (i.e.,~changing tank volume over time) and, further, a scenario in which pollution is entering and leaving a lake via streams.
\end{abstract}




\section*{SCENARIO DESCRIPTION}  
For this project, we will explore how the amount of drink powder in a tank changes when varying concentrations of powder and water solution are added to a tank that begins with pure water. As you perform these experiments, keep careful notes, recording any and all data and observations as well as answers to the questions in each part. You will need these to write your final lab report. To set up, stack one tank above the other as shown in Figure~\ref{fig:1-42-croppedIMG1362}, with a container under the bottom tank to catch the water flowing out.

\begin{figure}[ht]
	\centering
	% Requires \usepackage{graphicx}
	\includegraphics[width=5 true cm]{1-42-croppedIMG1362.jpg}
	
	\parbox{7 true cm}{\caption{\label{fig:1-42-croppedIMG1362} Set up for data collection.}}
\end{figure}



\subsection*{MATERIALS} 
The experiments should be completed in a lab environment with access to water (and paper towels!).
\begin{itemize}
	\item 2 drink dispensers (approximately 8 liters or larger)
	\item riser (for elevating one of the drink dispensers)
	\item collecting bucket
	\item balances (One for every two set-ups is sufficient.)
	\item bright-colored drink powder (Each group will need at least 100 grams.)
	\item timer (A cell phone app is sufficient.)
	\item pitcher (something to help fill the dispensers)
	\item mixing stick (Paint-mixing sticks work great.)
	\item tape and markers (if dispensers are not pre-marked)
\end{itemize}



\subsection*{Part I. Setting the stage}

For our experiments, we will assume that the flow rate out of the top tank and into the bottom tank is the same as the flow rate out of the bottom tank. We will also assume that the flow rate is constant throughout the experiments. If the flow rates are the same and constant, what should happen if the spouts on both tanks are opened fully at the same time? 

Let's first test that these assumptions are reasonable by determining the flow rate out of each tank; for this, you will need a timer, such as a stopwatch app. Fill both tanks with water to the same level, making note of the water level.  When your timer is ready, open both spouts simultaneously and watch what happens. Describe the results of this test and whether or not our assumptions are reasonable. Also calculate and make a note of what the flow rate is. Adjust the assumptions if necessary, and make notes about when the assumptions are valid (i.e.,~until the tank reaches a certain point) or any modifications you had to make to the process, e.g., holding one spout slightly more open. When you have finished performing this test, CLOSE THE SPOUTS and refill your tanks!

\subsection*{Part II. One spout open}

Now we will make a solution of water and drink powder in the top tank. We want this to be fairly concentrated, so use enough to make your mixture a nice vibrant color. Use the balances to carefully measure and record the amount of drink powder in grams that you are adding to the water. Make sure that ONLY the top tank contains the drink powder mixed with water. The bottom tank should have half the volume of the top tank, but initially be plain water. 
 
Before you let any liquids flow, describe what you expect to happen to the liquid in the bottom tank, which starts out as pure water. (Will it end as pure water? as powder-water mixture? Will it have the same concentration as the top tank? How long will it take for your predicted outcome to occur?) Sketch a graph of your prediction of the amount of powder-water mixture in the bottom tank at time $t$.
 
Let it flow! Open the spout on the top tank ONLY and watch what happens, recording any significant observations, including, but not limited to, timing how long you run your experiment. How long is a reasonable amount of time to conduct the experiment? Is there a limit on how long you can run the experiment?
 
Later you will create a model for the amount of drink powder $A$ in the bottom tank at time $t$, discuss what your model predicts in the short and long term, and use this model to determine the amount of drink powder (in grams) that is in the bottom tank at the end of your experiment. The differential equation that you start with should consider the rate of change of amount of drink powder, $\frac{dA}{dt}$. Write down any information NOW that you will need to write this model later. If time allows, you may work on this model before going on to the next part, but be sure to leave yourself enough time to do the next part. Note: For your model, you may assume that mixing is instantaneous.

\subsection*{Part III. Both spouts open}

This time we will again let the mixture flow from the top tank into the bottom tank, but we will start with plain water in the bottom tank of the SAME volume as the top tank. Carefully create your powder-water mixture, recording the amount of drink powder used in grams. Perform the experiment again, this time opening BOTH spouts simultaneously, and watch what happens. Make sure the collection container is in place below the bottom tank. How do you need to adjust your model for this scenario? How much drink powder $A$ do you think is in the second tank when the experiment ends? Is there a limit on how long you can run the experiment?

Modify your model to fit this second scenario and use it to answer the question of how much drink powder $A$ is in the second tank at the end of the experiment. Does your model predict what you expected? Discuss.

\subsection*{Part IV. Varying outflow rates}

(Modeling only; do not do this experiment.) At time $t=0$, a tank contains $A(0)=4$ grams (g) of drink powder dissolved in 100 liters (L) of water. Water containing 0.25 g of drink powder per liter is entering the tank at a rate of 3 L/minute, and the well-stirred solution leaves the tank at a rate of 3.5 L/minute.

\begin{itemize}
	\item [a)] Build a differential equation $\frac{dA}{dt}$ for the amount of drink powder $A$ in grams that is in the tank at time $t$ in minutes.
	\item [b)] Find an expression for the amount of drink powder, $A(t)$, in grams that is in the tank at time $t$ and plot $A(t)$ vs. $t$ over the time interval $[0, 200]$ minutes.
	\item [c)] Determine when the amount of drink powder doubles from the original amount in the tank.
	\item [d)] Determine when the amount of drink powder in the tank is 20 g.
	\item [e)] Determine when the amount of drink powder in the tank is 30 g.
	\item [f)] Determine the maximum amount of drink powder in the tank and when it occurs.
	\item [g)] Describe the long-term behavior of the amount of drink powder in the tank using accompanying plots to support your description.
\end{itemize}

\subsection*{Part V. Real-world application}

(Modeling only; do not do this experiment.) Mixing problems are often used to create models for pollution entering and leaving lakes through streams. Describe a scenario like this, and create a differential-equations model. State all assumptions and describe what your model predicts in the long term. You may consult outside resources to create a realistic model, but you should cite them. You may not take differential equations from outside sources without fully explaining them.

\section*{WRITE UP} 

Create a lab report that describes your models, assumptions, data, and solutions for each of the four scenarios (Parts II-V) in the following format. For each scenario give these sections:
\begin{enumerate}
	\item \textbf{Background}. Here you should include your assumptions, set up, and predictions and any changes you made to your assumptions after running the experiment.
	\item \textbf{The Model}. Here you should give the mathematical model, a justification for why this is an appropriate model, and a qualitative analysis of the differential equation (Examples are: What does the model predict in the long term? What are the equilibrium solutions?) Are there any restrictions on your model, i.e.,~is it only valid when $t>0$ or $A<1000$?
	\item \textbf{Solutions}. Give an analytic solution (if possible), plot the slope field with a graph of your solution on the slope field, and give a numerical approximation if an analytic solution is not available.
\end{enumerate}




\teacher{

\section*{NOTES FOR TEACHERS}

This activity provides students with a hands-on activity in which they perform experiments, then develop and solve equations that model each scenario. Their predictions and experimental results can be compared to the mathematical solutions. Further, the use of colored drink powder (i.e.,~the solution's color changes as the experiment progresses) reinforces what is happening experimentally. We have found that students show a better understanding of the mixing model after completing these experiments, as the modeled quantity (amount of drink powder in the bottom tank at any time $t$) is easier to keep in mind.

\subsection*{Part I.}

This set of experiments was completed in a 70-minute class period. As stated above, we assume that the flow rates into and out of the tanks are the same and constant. Realistically, slight adjustments, such as the student holding the spout open at a specific level, are necessary to ensure this is the case. Also, we have completed this set of experiments with two different capacity drink dispensers and have found that the smaller capacity (approximately 8 liters) saves time and yields the most effective results. Similarly, pre-marked volume ticks on the containers are another helpful time saver. Balances should be able to measure to $\pm$ 1 gram and handle at least 50 grams. Finally, while we have referred to our mixture formally as the ``powder-water mixture,'' in class we have verbally referred to it as the ``fruit drink.''


\subsection*{Part II.}

We begin with the mixing equation.

\begin{align*}
\dfrac{dA}{dt} &= rate_{in}-rate_{out} \\
\dfrac{dA}{dt} &= (C \cdot Q)_{in}-(C \cdot Q)_{out} \\
\dfrac{dA}{dt} &= (C \cdot Q)_{top}-(C \cdot Q)_{bottom} \\
\end{align*}

\noindent where $A$ is amount of drink powder in the bottom tank, $C$ is concentration of the powder-water mixture, $Q$ is flow rate of the powder-water mixture, and $t$ is time. (Note that different texts and teaching styles may choose to present the differential equation with simply amount per time and others may include the concentration and flow rate variables.)
 
In Part II, outflow rate is equal to zero because the spout on the bottom tank is closed. The equation reduces to:
\[ \dfrac{dA}{dt} = (C \cdot Q)_{top}\, .  \] 
Solving this differential equation using known values for top-tank concentration and outflow rate gives:
\[A(t) = [(C \cdot Q)_{top}] \cdot t + c   \]
where $c=0$ using the initial condition $A(0)=0$.
 
Using a bottom-tank initial volume of 3.8 L, a top-tank concentration of 8.56 g/L, and flow rate of 1.2 L/minute, our differential equation and solution are below. Note that these values are rounded.
\[ \dfrac{dA}{dt} = (8.56)(1.2) = 10.27,\quad  A(0)=0 \]
\[ A(t) = 10.27t \]
Based on our experiments, the slope field and plot of $A(t)$ vs. $t$ are presented in Figures~\ref{fig:1-42-Onespoutopenslopefield} and \ref{fig:1-42-Onespoutopenplot}.

\begin{figure}[!htb]
\centering
\begin{minipage}[b]{7 true cm}
\centering

\includegraphics[width=6 true cm]{1-42-Onespoutopenslopefield.pdf}
	
\parbox{7 true cm}{\caption{\label{fig:1-42-Onespoutopenslopefield} (Part II) Slope field of Drink Powder $A$ in g in bottom tank vs. Time in min.}}
\end{minipage}
\  \quad \
\begin{minipage}[b]{7 true cm}

\centering
\includegraphics[width=7 true cm]{1-42-Onespoutopenplot.pdf}
	\vspace*{.5 true cm}
	
\parbox{7 true cm}{\caption{\label{fig:1-42-Onespoutopenplot} (Part II) Plot of solution of Drink Powder $A$ in g in bottom tank as function of  Time in min.}}
\end{minipage}
\end{figure}



\subsection*{Part III.}
In this scenario, both spouts are open. Known quantities are top-tank concentration and outflow rate, bottom-tank volume ($V$) and outflow rate, and initial condition $A(0)=0$.

\begin{align*}
\dfrac{dA}{dt} &= (C \cdot Q)_{top}-(C \cdot Q)_{bottom} \\
\dfrac{dA}{dt} &= (C \cdot Q)_{top}-\left( \dfrac{A}{V} \cdot Q\right) _{bottom}
\end{align*} 

This results in a linear differential equation with solution: 
\[A(t)=\dfrac{(C \cdot Q)_{top}}{\left( \dfrac{Q}{V}\right) _{bottom}}+ce^{-t\left( \dfrac{Q}{V}\right) _{bottom}}\]
 
Since flow rates are assumed to be equal, this equation reduces to:
\[A(t)=(C_{top} \cdot V_{bottom})+c(C_{top} \cdot Q)e^{\dfrac{-Q\, t}{V_{bottom}}} \]
where
\[c=\dfrac{-V_{bottom}}{Q}\, .\]
 
Using a bottom-tank initial volume of 7.6 L, a top-tank concentration of 8.56 g/L, and flow rate of 1.2 L/minute, our differential equation is:
 
\[ \dfrac{dA}{dt} = (8.56)(1.2) - \frac{A}{7.6} (1.2) = 10.27-0.16A,\quad A(0)=0\, . \]
The \textit{Mathematica} solution to this differential equation is below, along with the plot of $A(t)$ vs. $t$ and the associated slope field. See Figures~\ref{fig:1-42-Bothspoutsopenslopefield} and \ref{fig:1-42-Bothspoutsopenplot}. Note that rounding will alter results slightly.  
\[ A(t) = 64.2\left( 1-e^{-0.16t}\right) \]



\begin{figure}[!htb]
\centering
\begin{minipage}[b]{7 true cm}
\centering

\includegraphics[width=6 true cm]{1-42-Bothspoutsopenslopefield.pdf}
	
\parbox{7 true cm}{\caption{\label{fig:1-42-Bothspoutsopenslopefield} (Part III) Slope field of Drink Powder $A$ in g in bottom tank vs. Time in min.}}
\end{minipage}
\  \quad \
\begin{minipage}[b]{7 true cm}

\centering
\includegraphics[width=7 true cm]{1-42-Bothspoutsopenplot.pdf}
	\vspace*{.5 true cm}
	
\parbox{7 true cm}{\caption{\label{fig:1-42-Bothspoutsopenplot} (Part III) Plot of solution of Drink Powder $A$ in g in bottom tank as function of  Time in min.}}
\end{minipage}
\end{figure}

We also suggest that Part III could be extended to consider the collection container to be a third tank.  

\subsection*{Part IV.} 
This is a standard mixing problem involving differing flow rates, although the difficulty of this part can be adjusted for each group of students. Our experience is that this part helps students connect what they learned experimentally in the lab to more abstract modeling situations that they will encounter in homework and on tests. The resulting mathematical model is: 
\[ \dfrac{dA}{dt} = 0.75-\dfrac{3.5A}{(100-0.5t)}, \hspace{1in} A(0)=4\, . \]
The solution is below.
\[A(t)=0.25(100-0.5t)-(2.1 \times 10^{-13})(100-0.5t)^7\]
 
The slope field and plot of $A(t)$ vs. $t$ are shown in in Figures~\ref{fig:1-42-Standardslopefield} and \ref{fig:1-42-Standardplot}. 


\begin{figure}[!htb]
\centering
\begin{minipage}[b]{7 true cm}
\centering

\includegraphics[width=6 true cm]{1-42-Standardslopefield.pdf}
	
\parbox{7 true cm}{\caption{\label{fig:1-42-Standardslopefield} (Part IV) Slope field of Drink Powder $A$ in g in bottom tank vs. Time in min.}}
\end{minipage}
\  \quad \
\begin{minipage}[b]{7 true cm}

\centering
\includegraphics[width=7 true cm]{1-42-Standardplot.pdf}
	\vspace*{.5 true cm}
	
\parbox{7 true cm}{\caption{\label{fig:1-42-Standardplot} (Part IV) Plot of solution of Drink Powder $A$ in g in bottom tank as function of  Time in min.}}
\end{minipage}
\end{figure}

\vspace*{-.2 true cm}
\subsection*{Part V.} 

Students should explore different situations in which pollution may be affecting a lake. Since this part is very open ended, we hope students will think outside the box. As long as reasonable and justified models are created and any outside resources are well cited, full credit will be given.

}
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