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Exploring Calculus with Maple Introductory Calculus

Stephen Brown, M.Sc., Lindsey Reinholz, Ph.D. (Editor)

Section 3.14 Tank Mixing Problem

Subsection 3.14.1 Recommended Tutorials

Before starting on these exercises, you should familiarize yourself with the material covered in the following tutorials:
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Subsection 3.14.2 Introduction

In this activity, you will investigate differential equations that arise from the mixture of fluid in a tank. You will need to have a basic understanding of the relationship between mass, volume, and concentration.
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Concentration is defined as
\begin{equation*} \text{ concentration } =\frac{\text{ mass } }{\text{ volume } }\text{.} \end{equation*}
Therefore, to determine mass given a volume and concentration,
\begin{equation*} \text{ mass } =\text{ concentration } \times\text{ volume }\text{.} \end{equation*}
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Suppose you are having a wedding and you start with a 5 L tank of coffee that has a concentration of 60 g/L. The wedding guests are drinking the coffee at a rate of 0.2 L/min. You are refilling the tank at a rate of 0.15 L/min with coffee that has a concentration of 50 g/L.
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Figure 3.9. An illustration of the tank of coffee. The initial conditions for the volume and mass of coffee are shown.
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In the following exercises, you will set up and solve differential equations that describe the mass of coffee in the tank as well as the concentration of coffee in the tank at a given time.
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Exercises 3.14.3 Exercises

1.

To begin, you will need to determine the volume of coffee in the tank at any given time. Since the flow rate in is different from the flow rate out, this volume is not constant.
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(a)
Set up a differential equation for the rate of change of volume of coffee in the tank using \(V'(t) = (\text{ rate in } ) - (\text{ rate out } )\text{.}\)
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(b)
Solve this differential equation using the initial condition to obtain a formula for the volume \(V(t)\) after \(t\) minutes.
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(c)
Use the solve() command to compute how long it will be until you run out of coffee.
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2.

Using the given information and the volume of coffee in the tank at time \(t\) from the previous exercise, you can now determine the mass of coffee in the tank at any given time.
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(a)
Set up a differential equation for the mass of coffee in the tank using
\begin{equation*} m'(t) = \lrp{\begin{array}{l} \text{ concentration } \\ \text{ entering tank } \end{array} } \lrp{\begin{array}{l} \text{ volume } \\ \text{ rate in } \end{array} } - \lrp{\begin{array}{l} \text{ concentration } \\ \text{ in tank at time } t \end{array} } \lrp{\begin{array}{l} \text{ volume } \\ \text{ rate out } \end{array} }\text{.} \end{equation*}
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Hint.
The concentration entering the tank, the volume rate in, and the volume rate out are all given. However, the concentration in the tank at time \(t\) will need to be given as a fraction of mass \(m(t)\) divided by volume \(V(t)\text{.}\) This volume comes from the previous exercise, but \(m(t)\) remains an unknown function until you can solve this differential equation.
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(b)
Solve the differential equation for mass using your initial condition for the mass of the coffee \(m(t)\) after \(t\) minutes.
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3.

Determine the function for the concentration of the coffee in the tank \(C(t)\) after \(t\) minutes using \(C(t) = m(t)/V(t)\text{.}\)
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4.

Plot \(V(t)\text{,}\) \(m(t)\text{,}\) and \(C(t)\) on separate graphs to see what happens to the mass, volume, and concentration over time.
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5.

Use the limit() command to determine the concentration of coffee as you approach the last drop of coffee.
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