2. Nomenclature

In this section we introduce a specific terminology useful to allow understanding of the terms properly.

In order to analyze MPs and MP-matrices, we begin by studying a problem already considered in a previous book chapter [6], which involves a tank with five compartments, shown in Figure 1.

In this scheme, C<sup>0</sup> is the initial concentration (e.g., salt concentration in water at the entrance of the tank system), Ci is the concentration in the ith compartment (i = 1,…,5), and Φ<sup>0</sup> 6¼ 0 is the incoming and also outgoing flux.

For instance, if Φ1<sup>k</sup> is the flux that goes from the left (first) to the kth compartment (being k = 2, 3, 4) and V<sup>1</sup> is the volume of the first container, then a mass balance gives the following ODE:

$$V\_1 \frac{dC\_1}{dt} = \Phi\_0 C\_0 - \Phi\_{12} C\_1 - \Phi\_{13} C\_1 - \Phi\_{14} C\_1 = \Phi\_0 C\_0 - \left(\Phi\_{12} + \Phi\_{13} + \Phi\_{14}\right) C\_1 \tag{1}$$

The ODEs associated with the central compartments (i = 2, 3, 4) are simpler, since in each case, there is only one incoming flux Φ1<sup>k</sup> (being k = 2, 3, 4) and a unique outgoing flux Φk<sup>5</sup> (being

Figure 1. A tank with five internal compartments.

In particular, the ODE system-associated matrix deserves to be studied since it determines the

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

Taking into account previous results about MP-matrices, and the new ones presented here, two

• If the MP corresponds to an open system, then the solutions are asymptotically stable.

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

In this section we introduce a specific terminology useful to allow understanding of the terms

In order to analyze MPs and MP-matrices, we begin by studying a problem already considered in a previous book chapter [6], which involves a tank with five compartments, shown in

In this scheme, C<sup>0</sup> is the initial concentration (e.g., salt concentration in water at the entrance of the tank system), Ci is the concentration in the ith compartment (i = 1,…,5), and Φ<sup>0</sup> 6¼ 0 is the

For instance, if Φ1<sup>k</sup> is the flux that goes from the left (first) to the kth compartment (being k = 2, 3, 4) and V<sup>1</sup> is the volume of the first container, then a mass balance gives the following ODE:

The ODEs associated with the central compartments (i = 2, 3, 4) are simpler, since in each case, there is only one incoming flux Φ1<sup>k</sup> (being k = 2, 3, 4) and a unique outgoing flux Φk<sup>5</sup> (being

ð1Þ

qualitative behavior of the solutions.

42 Matrix Theory-Applications and Theorems

of the associated matrices, so-called MP-matrix.

• All the solutions of a given MP are stable.

models a given MP) should be deeply analyzed.

main conjectures can be proposed:

2. Nomenclature

incoming and also outgoing flux.

properly.

Figure 1.

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

$$V\_2 \frac{d\mathbb{C}\_2}{dt} = \Phi\_{12}\mathbb{C}\_1 - \Phi\_{25}\mathbb{C}\_2,\\ V\_3 \frac{d\mathbb{C}\_3}{dt} = \Phi\_{13}\mathbb{C}\_1 - \Phi\_{35}\mathbb{C}\_3,\\ V\_4 \frac{d\mathbb{C}\_4}{dt} = \Phi\_{14}\mathbb{C}\_1 - \Phi\_{45}\mathbb{C}\_4 \tag{2}$$

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

$$V\_{\varsigma} \frac{dC\_{\varsigma}}{dt} = \Phi\_{2\varsigma} C\_2 + \Phi\_{3\varsigma} C\_3 + \Phi\_{4\varsigma} C\_4 - \Phi\_0 C\_9 \tag{3}$$

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

$$\begin{cases} V\_1 \frac{d\mathbf{C\_1}}{dt} = \Phi\_0 \mathbf{C\_0} - (\Phi\_{12} + \Phi\_{13} + \Phi\_{14}) \mathbf{C\_1} \\\\ V\_2 \frac{d\mathbf{C\_2}}{dt} = \Phi\_{12} \mathbf{C\_1} - \Phi\_{25} \mathbf{C\_2} \\\\ V\_3 \frac{d\mathbf{C\_3}}{dt} = \Phi\_{13} \mathbf{C\_1} - \Phi\_{35} \mathbf{C\_3} \\\\ V\_4 \frac{d\mathbf{C\_4}}{dt} = \Phi\_{14} \mathbf{C\_1} - \Phi\_{45} \mathbf{C\_4} \\\\ V\_5 \frac{d\mathbf{C\_5}}{dt} = \Phi\_{25} \mathbf{C\_2} + \Phi\_{35} \mathbf{C\_3} + \Phi\_{45} \mathbf{C\_4} - \Phi\_0 \mathbf{C\_5} \end{cases} (4)$$

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

$$\mathbf{C} = \begin{pmatrix} \mathbf{C}\_1 \\ \mathbf{C}\_2 \\ \mathbf{C}\_3 \\ \mathbf{C}\_4 \\ \mathbf{C}\_5 \end{pmatrix} \text{ and } \mathbf{B} = \begin{pmatrix} \Phi\_0/V\_1 \\ 0 \\ 0 \\ 0 \\ 0 \end{pmatrix} \tag{5}$$

The system-associated matrix (MP-matrix) is

$$\mathbf{A} = \begin{pmatrix} -(\Phi\_{12} + \Phi\_{13} + \Phi\_{14})/V\_1 & 0 & 0 & 0 & 0 \\ & \Phi\_{12}/V\_2 & -\Phi\_{25}/V\_2 & 0 & 0 & 0 \\ & \Phi\_{13}/V\_3 & 0 & -\Phi\_{35}/V\_3 & 0 & 0 \\ & \Phi\_{14}/V\_4 & 0 & 0 & -\Phi\_{45}/V\_4 & 0 \\ & 0 & \Phi\_{25}/V\_5 & \Phi\_{35}/V\_5 & \Phi\_{45}/V\_5 & -\Phi\_0/V\_5 \end{pmatrix} \tag{6}$$

Finally, it is important to observe that in both examples (Figures 1 and <sup>2</sup>), we have <sup>P</sup>Φ<sup>i</sup> <sup>¼</sup> <sup>P</sup>Φk, being Φ<sup>i</sup> all the system incoming fluxes and Φ<sup>k</sup> the corresponding outgoing fluxes. This equation must be satisfied, since the compartments are neither filled up nor emptied with time, at least for

Square Matrices Associated to Mixing Problems ODE Systems

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

45

In that case all the compartment volumes remain constant, and so if in an MP the following equation <sup>P</sup>Φ<sup>i</sup> <sup>¼</sup> <sup>P</sup>Φ<sup>k</sup> (being <sup>Φ</sup><sup>i</sup> all the system incoming fluxes and <sup>Φ</sup><sup>k</sup> the corresponding outgoing fluxes) is satisfied, it will refer to a mixing problem with constant volumes (MP-CV). Taking into account all these terms, several previous results can be reformulated, as shown in

In order to give some general results, it is convenient to consider two different situations: MP

Considering again the example in Figure 1, it is possible to enumerate the compartments, such that the flux always goes from the ith container to the jth one, being i < j, shown in brackets. Analyzing the system (Eq. (4)), it is easy to observe that for the jth container, the ODE right hand side is a linear combination of a subset of , and this result can be extended straightforward. In fact, in a previous book chapter [6], it was proved that if in a given MP the compartments can be enumerated such that there is no recirculation (i.e., if there is no flux from compartment to compartment ), then the ODE corresponding to the jth

As a consequence, under the previous conditions, the corresponding ODE system has an

<sup>C</sup><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> V1

C1

Revisiting the ODE system (Eq. (4)), corresponding to Figure 1, it can be rewritten as

<sup>C</sup><sup>1</sup> � <sup>Φ</sup><sup>25</sup> V2 C2

<sup>C</sup><sup>1</sup> � <sup>Φ</sup><sup>35</sup> V3 C3

<sup>C</sup><sup>1</sup> � <sup>Φ</sup><sup>45</sup> V4 C4

> Φ<sup>35</sup> V5 C<sup>3</sup> þ

Φ<sup>45</sup> V5

<sup>C</sup><sup>4</sup> � <sup>Φ</sup><sup>0</sup> V5 C5 ð7Þ

(8)

the typical MPs' real-life most interesting situations.

the next section.

3. Previous results revisited

compartment will be of the form:

associated upper matrix.

being and

dC<sup>1</sup> dt <sup>¼</sup> <sup>Φ</sup><sup>0</sup> V1

8

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

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

dC<sup>2</sup> dt <sup>¼</sup> <sup>Φ</sup><sup>12</sup> V2

dC<sup>3</sup> dt <sup>¼</sup> <sup>Φ</sup><sup>13</sup> V3

dC<sup>4</sup> dt <sup>¼</sup> <sup>Φ</sup><sup>14</sup> V4

dC<sup>5</sup> dt <sup>¼</sup> <sup>Φ</sup><sup>25</sup> V5 C<sup>2</sup> þ

without recirculation and MP with recirculation.

Hereafter, we will call MP-matrix to any ODE system-associated matrix related to a given MP, like matrix A of Eq. (6).

In the previous example, the MP-matrix obviously depends on the numbers given to the different containers. In that example it was possible to enumerate the compartments such that the flux always goes from the ith compartment to the jth one, where . For instance, a possible enumeration for this purpose is the one illustrated in Figure 1.

In general, if in a given MP it is possible to enumerate the containers such that the flux always goes from the ith compartment to the jth one, with , then the MP will be considered as a mixing problem without recirculation (MP-WR).

Now, let us analyze a different problem, where a couple of tanks are linked by all possible connections between them, including recirculation from the second tank back to the first one, as in Figure 2. This problem represents an interesting variation of an MP analyzed by Zill [7] in his textbook, where the main difference is that this new MP has no incoming and/or outgoing flux, i.e., it is a closed system.

If in a given MP we have that <sup>P</sup>Φ<sup>i</sup> <sup>¼</sup> 0, being <sup>Φ</sup><sup>i</sup> all the system incoming fluxes, and <sup>P</sup>Φ<sup>k</sup> <sup>¼</sup> 0, being Φ<sup>k</sup> all the system outgoing fluxes, then it will be named MP closed system (MP-CS). Otherwise, it will be an open system (MP-OS).

Taking into account the abovementioned nomenclature, the example considered in Figure 2 corresponds to an MP-CS, while the MP analyzed in Zill's textbook [7] is an MP-OS, and both are systems with recirculation.

Figure 2. Two tanks with recirculation and no incoming or outgoing fluxes.

Finally, it is important to observe that in both examples (Figures 1 and <sup>2</sup>), we have <sup>P</sup>Φ<sup>i</sup> <sup>¼</sup> <sup>P</sup>Φk, being Φ<sup>i</sup> all the system incoming fluxes and Φ<sup>k</sup> the corresponding outgoing fluxes. This equation must be satisfied, since the compartments are neither filled up nor emptied with time, at least for the typical MPs' real-life most interesting situations.

In that case all the compartment volumes remain constant, and so if in an MP the following equation <sup>P</sup>Φ<sup>i</sup> <sup>¼</sup> <sup>P</sup>Φ<sup>k</sup> (being <sup>Φ</sup><sup>i</sup> all the system incoming fluxes and <sup>Φ</sup><sup>k</sup> the corresponding outgoing fluxes) is satisfied, it will refer to a mixing problem with constant volumes (MP-CV).

Taking into account all these terms, several previous results can be reformulated, as shown in the next section.
