Compartment Model

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 containinggallons of solute.Pure water flows at the raterinto the first tank and is mixed, then flows at the raterinto the second tank and is mixed, then flows at the raterinto thethird tank and is mixed, and finally flows out of the third tank at the rater.We define the constants.Then the differential equation for the system is

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



with the initial conditions

.

Plot the solution curves forfor.

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 containinggallons of solute.The liquid flows at the raterfrom the first tank and is mixed, then flows at the raterinto the second tank and is mixed, then flows at the raterinto thethird 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 forfor.

The Compartment Model III.

The "five-stage system" consists of five tanks containinggallons 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 forfor.