The Compartment Model I.

    The "compartment" model is used to describe the concentration of a dissolved substance in several compartments in a system.  For example, the "three-stage system" consists of three tanks containing    gallons of solute.  Pure water flows at the rate  r  into the first tank and is mixed, then flows at the rate  r  into the second tank and is mixed, then flows at the rate  r  into the  third tank and is mixed, and finally flows out of the third tank at the rate  r.  We define the constants  .  Then the differential equation for the system is

      

Proof  The Compartment Model  The Compartment Model  

Computer Programs  The Compartment Model  The Compartment Model  

Example 1.  Find the solution to the "three-stage system" where  

    

with the initial conditions  

    .

Plot the solution curves for    for  .
Solution 1.

The Compartment Model II.

    Another "compartment" model is used to describe the concentration of a dissolved substance in several compartments in a system.  For example, the "closed" three-stage system consists of three tanks containing    gallons of solute.  The liquid flows at the rate  r  from the first tank and is mixed, then flows at the rate  r  into the second tank and is mixed, then flows at the rate  r  into the  third tank and is mixed, and finally flows out of the third tank and back into the first tank at the rate r.  We define the constants  .  Then the differential equation for the system is

      

Example 2.  Find the solution to the "closed" three-stage system where  

    

with the initial conditions  

    .

Plot the solution curves for    for  .  
Solution 2.

The Compartment Model III.

    The "five-stage system" consists of five tanks containing    gallons of solute.  Pure water flows at the rate into the first tank and is mixed, then flows at the rate into the second tank and is mixed, then flows at the rate into successive tanks and is mixed, and finally flows out of the fifth tank at the rate .  
We define the constants  , , , , and  .  Then the differential equation for the system is

        

with the initial conditions    

    

Remark. For a tractable example, if the size of the tanks is increasing then ordinary eigenvectors are obtained to form the solution.  Otherwise, the system might require "generalized eigenvectors" which are important, but we do not want to digress in that direction in this module.

Example 3.  Find the solution to the "five-stage system" where  

       

with the initial conditions   

    .  

Plot the solution curves for    for  .  
Solution 3.

Old Lab Project (The Compartment Model  The Compartment Model).  Internet hyperlinks to an old lab project.  

Research Experience for Undergraduates

The Compartment Model  The Compartment Model  Internet hyperlinks to web sites and a bibliography of articles.  

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(c) John H. Mathews 2004