Monday, February 1, 2010

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 containing [Graphics:Images/CompartmentModelMod_gr_1.gif] 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 [Graphics:Images/CompartmentModelMod_gr_2.gif]. Then the differential equation for the system is

[Graphics:Images/CompartmentModelMod_gr_3.gif]

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 [Graphics:Images/CompartmentModelMod_gr_90.gif] 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 [Graphics:Images/CompartmentModelMod_gr_91.gif]. Then the differential equation for the system is

[Graphics:Images/CompartmentModelMod_gr_92.gif]

The Compartment Model III:

The "five-stage system" consists of five tanks containing [Graphics:Images/CompartmentModelMod_gr_215.gif] gallons of solute. Pure water flows at the rate [Graphics:Images/CompartmentModelMod_gr_216.gif] into the first tank and is mixed, then flows at the rate [Graphics:Images/CompartmentModelMod_gr_217.gif] into the second tank and is mixed, then flows at the rate [Graphics:Images/CompartmentModelMod_gr_218.gif] into successive tanks and is mixed, and finally flows out of the fifth tank at the rate [Graphics:Images/CompartmentModelMod_gr_219.gif].
We define the constants [Graphics:Images/CompartmentModelMod_gr_220.gif], [Graphics:Images/CompartmentModelMod_gr_221.gif], [Graphics:Images/CompartmentModelMod_gr_222.gif], [Graphics:Images/CompartmentModelMod_gr_223.gif], and [Graphics:Images/CompartmentModelMod_gr_224.gif]. Then the differential equation for the system is

[Graphics:Images/CompartmentModelMod_gr_225.gif]

with the initial conditions

[Graphics:Images/CompartmentModelMod_gr_226.gif]

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.

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Introductory Methods of Numerical Analysis