1. Introduction

Mixing problems (MPs), also known as "compartment analysis" [1], in chemistry involve creating a mixture of two or more substances and then determining some quantity (usually concentration) of the resulting mixture. For instance, a typical mixing problem deals with the amount of salt in a mixing tank. Salt and water enter to the tank at a certain rate, they are mixed with what is already in the tank, and the mixture leaves at a certain rate. This process is modeled by an ordinary differential equation (ODE), as Groestch affirms: "The direct problem for one-compartment mixing models is treated in almost all elementary differential equations texts" [2].

Instead of only one tank, there is a group, as it was stated by Groestch: "The multicompartment model is more challenging and requires the use of techniques of linear algebra" [2].

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In particular, the ODE system-associated matrix deserves to be studied since it determines the qualitative behavior of the solutions.

In several previous papers and book chapters [3–6], MPs were studied from different points of view. In the first paper [3], a particular MP with three compartments was proposed, and after applying Laplace transform, this example was connected with important concepts in reactor design, like the transference function. 2 years later, another work [4] analyzed more general MPs in order to obtain characterization results independent of the internal geometry of the tank system. In the third paper [5], the educative potential of MPs was studied, focusing on inverse modeling problems. Finally, in a recent book chapter [6], results for MPs with and without recirculation of fluids were analyzed, and other general results were obtained.

In all these works, a given MP is modeled through an ODE linear system, in which qualitative properties (like stability and asymptotic stability) depend on the eigenvalues and eigenvectors of the associated matrices, so-called MP-matrix.

k = 2, 3, 4). Once again, if Vk is the volume of the kth container, these equations can be written

dt <sup>¼</sup> <sup>Φ</sup>13C<sup>1</sup> � <sup>Φ</sup>35C3, V<sup>4</sup>

dt <sup>¼</sup> <sup>Φ</sup>0C<sup>0</sup> � ð Þ <sup>Φ</sup><sup>12</sup> <sup>þ</sup> <sup>Φ</sup><sup>13</sup> <sup>þ</sup> <sup>Φ</sup><sup>14</sup> <sup>C</sup><sup>1</sup>

dt <sup>¼</sup> <sup>Φ</sup>25C<sup>2</sup> <sup>þ</sup> <sup>Φ</sup>35C<sup>3</sup> <sup>þ</sup> <sup>Φ</sup>45C<sup>4</sup> � <sup>Φ</sup>0C<sup>5</sup>

1

CCCCCCA

0

BBBBBB@

After some algebraic manipulations, the corresponding mathematical model can be written as

and B ¼

dC<sup>4</sup>

Square Matrices Associated to Mixing Problems ODE Systems

http://dx.doi.org/10.5772/intechopen.74437

dt <sup>¼</sup> <sup>Φ</sup>14C<sup>1</sup> � <sup>Φ</sup>45C<sup>4</sup> (2)

ð3Þ

43

(4)

(5)

dC<sup>3</sup>

If all these equations are put together, the following ODE system is obtained:

dt <sup>¼</sup> <sup>Φ</sup>12C<sup>1</sup> � <sup>Φ</sup>25C<sup>2</sup>

dt <sup>¼</sup> <sup>Φ</sup>13C<sup>1</sup> � <sup>Φ</sup>35C<sup>3</sup>

dt <sup>¼</sup> <sup>Φ</sup>14C<sup>1</sup> � <sup>Φ</sup>45C<sup>4</sup>

as.

V2 dC<sup>2</sup>

dt <sup>¼</sup> <sup>Φ</sup>12C<sup>1</sup> � <sup>Φ</sup>25C2, V<sup>3</sup>

V1 dC<sup>1</sup>

8

>>>>>>>>>>>>>><

>>>>>>>>>>>>>>:

, where.

V2 dC<sup>2</sup>

V3 dC<sup>3</sup>

V4 dC<sup>4</sup>

V5 dC<sup>5</sup>

C ¼

C1 C2 C3 C4 C5 1

CCCCCCA

0

BBBBBB@

Finally, for the right (fifth) container, we have:

Figure 1. A tank with five internal compartments.

Taking into account previous results about MP-matrices, and the new ones presented here, two main conjectures can be proposed:


In order to investigate if these conjectures—among others, introduced in the following sections —are true or not, MP-matrices (i.e., square matrices associated to the ODE linear system that models a given MP) should be deeply analyzed.
