Differential Equations Day 12 -- Project 3 -- Mixing Models
Directions
Follow the steps below and do your work in the provided GeoGebra windows.
NOTE to VSC students taking this class with me: be sure you are accessing this through a GeoGebra Classroom Link! The URL of this page should have the word “classroom” in it. If not, then go back to Canvas and be sure to access this page from the Project 3 page. Also: I strongly recommend you login to GeoGebra.org with a free account so your work is saved, and you can come back later to review or modify it.
For external readers: these "project" activities are meant to be taken as part of my course, so these sections of the GeoGebra book my not be as intelligible as others. My apologies.
Goals
The primary goal of this project is to learn about Mixing Models of chemical solutions. A secondary goal is that you will also continue to gain experience with algebraic and numerical methods.
Content Overview
Mixing models are a classic application of first order (and, in fact, higher order) differential equations. They are used to make a dynamic model of the amount of some substance dissolved in a chemical solution. The substance can be anything that dissolves in a well-mixed liquid. We'll stick to salt dissolved in water in this project.
All mixing models are governed by the mixing principle
The rate of change of a substance dissolved in a solution in a tank is equal to the amount of the substance being added to the tank less the amount of the substance being removed from the tank.The mixing principle is not a complex principle, but we'll see it leads to interesting differential equation models with interpretable solutions. We can put the mixing principle into the form of a first order differential equation: We'll explore how to engage more meaningfully with
RateSubstanceIn and RateSubstanceOut in the parts below. Steps For Part 1
The Setup: A large tank holds 300 gallons, and contains a mixing turbine that keeps any solution well mixed in the tank. A solution of salt and water is pumped into the tank at 3 gallons per minute; 2 pounds of salt are dissolved in each gallon of the incoming solution. As the incoming solution is added, 3 gallons per minute of the well mixed solution in the tank is pumped out. Suppose the 300 gallon tank initially holds a solution with 50 total pounds of salt suspended.
- Declare a function of two variables:
RateSubstanceIn(x,y)=2*3 - Declare
RateSubstanceOut(x,y)=y/300*3 - What are the units of these functions? Make a note in a textbox.
- Declare
f(x,y)=RateSubstanceIn-RateSubstanceOut - Plot the slope field for
f - Plot the initial condition
InitialCondition=(0,50) - Use Locus to get a numerical integral estimate of the specific solution of the differential equation passing through
InitialCondition - Use an appropriate algebraic method to find the specific solution of the differential equation that matches
InitialCondition. Plot your findings asSpecificSolution(x) - Plot the stable solution of the differential equation as a line
y=#where you should replace # with a specific number. You can determine this number either by contemplating the setup, or by examining the algebraic solution of the differential equation, or some other means. The choice is yours. - Use either numerical methods or your algebraic solution (whichever you think is best) to determine how long it will take (in minutes) before the salt mixture will be within 45 pounds of salt of the stable solution.
Part 1 Applet
Part 2 (not broken out by steps)
The Setup: A large tank is partially filled with 100 gallons of fluid in which 10 pounds of salt is dissolved. Brine consisting of 1/2 pound of salt per gallon is pumped into the tank at a rate of 6 gallons per minute. The will mixed solution is then pumped out at a slower rate of 4 gallons per minute. Find the number of pounds of salt in the tank after 30 minutes. Plot your results in the GeoGebra applet below. You may use either numerical or algebraic methods.