4. Some considerations about terminology

We start this section explaining three simple and intuitive terms.

Firstly, we will consider that an input tank is a tank with one or more incoming fluxes. Secondly, a tank with one or more outgoing fluxes will be called output tank. Finally, we will say that an internal tank is a tank without incoming and/or outgoing fluxes to or from outside the system.

Taking into account the previous nomenclature, if Φki ∀k ¼ 1, 2, ⋯, m represent all the ith tank incoming fluxes, then <sup>P</sup>Φki 6¼ 0 for an input tank, and in the same way, if <sup>Φ</sup>jk <sup>∀</sup><sup>k</sup> <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, <sup>⋯</sup>, m represent all the jth tank outgoing fluxes, then <sup>P</sup>Φjk 6¼ 0 for an output tank.

Input and output tanks are not mutually exclusive. For instance, in Figure 4, the first tank is an input tank, and at same time, it is an output tank, since it has an incoming flux Φ<sup>0</sup> from outside

Figure 4. A tank system with recirculation and with incoming and outgoing fluxes.

Now, let us consider again the MP-matrix , corresponding to the system of Figure 2:

0

BB@

� <sup>Φ</sup><sup>12</sup> V1

> Φ<sup>12</sup> V2

� <sup>Φ</sup><sup>12</sup> V1 þ ε

> Φ<sup>12</sup> V2

It is easy to observe that this new matrix will not satisfy the condition given by Eq. (14). Moreover, there is no MP associated to this matrix , since this condition must be satisfied

As a first consequence, not every square matrix is an MP-matrix. A second observation is that if

Furthermore, if volumes and fluxes are multiplied by a scale factor, then the MP-matrix Eq. (11) remains unchanged, and so, a scale factor in geometry, not in concentrations, produces

After interpreting the previous results, we note that when working with MP-matrices, existence, uniqueness, and stability questions for the inverse-modeling problem have negative

The same situation can be observed in many other inverse problems [2], and it is not an

Firstly, we will consider that an input tank is a tank with one or more incoming fluxes. Secondly, a tank with one or more outgoing fluxes will be called output tank. Finally, we will say that an internal tank is a tank without incoming and/or outgoing fluxes to or from outside the system.

Taking into account the previous nomenclature, if Φki ∀k ¼ 1, 2, ⋯, m represent all the ith tank incoming fluxes, then <sup>P</sup>Φki 6¼ 0 for an input tank, and in the same way, if <sup>Φ</sup>jk <sup>∀</sup><sup>k</sup> <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, <sup>⋯</sup>, m

Input and output tanks are not mutually exclusive. For instance, in Figure 4, the first tank is an input tank, and at same time, it is an output tank, since it has an incoming flux Φ<sup>0</sup> from outside

represent all the jth tank outgoing fluxes, then <sup>P</sup>Φjk 6¼ 0 for an output tank.

a given MP-matrix is slightly changed, the result is not necessarily a new MP-matrix.

Φ<sup>21</sup> V1

1

CCA

Φ<sup>21</sup> V1

1

CCA

� <sup>Φ</sup><sup>21</sup> V2

(15)

(16)

� <sup>Φ</sup><sup>21</sup> V2

A ¼

If is slightly changed only in its first entry, we have the following matrix:

0

BB@

A<sup>ε</sup> ¼

independently of the internal geometry of the system.

exactly the same mathematical model.

48 Matrix Theory-Applications and Theorems

exclusive property of compartment analysis.

4. Some considerations about terminology

We start this section explaining three simple and intuitive terms.

answers.

the system and it also has an outgoing flux Φ<sup>0</sup> that leaves the tank system. It should be noted that in Figure 4, the second tank is an internal one.

Another interesting example was proposed by Boelkins et al. [8]. The authors considered a three-tank system connected such that each tank contains an independent inflow that drops salt solution to it, each individual tank has a separated outflow, and each one is connected to the rest of them with inflow and outflow pipes. In this case, all tanks are input and output ones, and there is no internal tank.

It is important to mention that those types of tanks or compartments play different roles in the ODE-associated system and also—as a consequence—in the corresponding MP-matrix. In order to show this fact, let us examine a three-tank system with all the possible connections among them, as in Figure 5.

As a first remark, Figure 5 system has recirculation—unless Φ<sup>21</sup> ¼ Φ<sup>32</sup> ¼ Φ<sup>31</sup> ¼ 0, which represents a trivial case—and consequently, an associated upper MP-matrix will not be expected for this problem.

In the mass balance for the first tank—which is an input one—a term Φ0C<sup>0</sup> must be considered. In the same way, in the mass balance of the third tank—which is an output one—a term Φ0C<sup>3</sup> will appear. These two terms will not be part of the second equation of the ODE system, which can be formulated as follows:

$$\begin{cases} V\_1 \frac{dC\_1}{dt} = \Phi\_0 C\_0 + \Phi\_{21} C\_2 + \Phi\_{31} C\_3 - \left(\Phi\_{12} + \Phi\_{13}\right) C\_1 \\ V\_2 \frac{dC\_2}{dt} = \Phi\_{12} C\_1 + \Phi\_{32} C\_3 - \left(\Phi\_{21} + \Phi\_{23}\right) C\_2 \\ V\_3 \frac{dC\_3}{dt} = \Phi\_{13} C\_1 + \Phi\_{23} C\_2 - \left(\Phi\_{31} + \Phi\_{32} + \Phi\_0\right) C\_3 \end{cases} \tag{17}$$

Figure 5. Three tanks with all the possible connections.

Once again, the ODE system can be written as , where the MP-matrix is:

$$\mathbf{A} = \begin{pmatrix} \left( -1/V\_1 \right) \left( \Phi\_{12} + \Phi\_{13} \right) & \Phi\_{21}/V\_1 & \Phi\_{31}/V\_1 \\ \Phi\_{12}/V\_2 & \left( -1/V\_2 \right) \left( \Phi\_{21} + \Phi\_{23} \right) & \Phi\_{32}/V\_2 \\ \Phi\_{13}/V\_3 & \Phi\_{23}/V\_3 & \left( -1/V\_3 \right) \left( \Phi\_{31} + \Phi\_{32} + \Phi\_0 \right) \end{pmatrix} \tag{18}$$

In the previous ODE system, the independent vector is:

$$\mathbf{B} = \begin{pmatrix} \Phi\_0 \\ \vdots \\ \mathbf{0} \\ \mathbf{0} \\ \mathbf{0} \end{pmatrix} \tag{19}$$

Theorem (Gershgorin)

compartment considered.

dt <sup>¼</sup> <sup>Φ</sup><sup>0</sup>

<sup>V</sup><sup>1</sup> <sup>C</sup><sup>0</sup> <sup>þ</sup> <sup>Φ</sup><sup>21</sup>

look like the one schematized in Figure 6.

<sup>V</sup><sup>2</sup> <sup>C</sup><sup>3</sup> � ð Þ <sup>Φ</sup>21þΦ<sup>23</sup>

<sup>V</sup><sup>2</sup> C2.

Φ<sup>32</sup> V2

Figure 6. The Gershgorin disk corresponding to an input tank.

matrix (Eq. (18)) that can be written as � ð Þ <sup>Φ</sup><sup>12</sup> <sup>þ</sup> <sup>Φ</sup><sup>13</sup>

expressed as dC<sup>1</sup>

<sup>R</sup><sup>1</sup> <sup>¼</sup> <sup>Φ</sup>21þΦ<sup>31</sup> <sup>V</sup><sup>1</sup> .

<sup>V</sup><sup>2</sup> <sup>C</sup><sup>1</sup> <sup>þ</sup> <sup>Φ</sup><sup>32</sup>

� ð Þ <sup>Φ</sup><sup>21</sup> <sup>þ</sup> <sup>Φ</sup><sup>23</sup> V2

.

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

> Φ<sup>12</sup> V2

If is an matrix, with entries being , and is the sum of

the non-diagonal entry modules in the th row, then every eigenvalue of lies within at least

This theorem was widely used in previous book chapters [6, 10, 11] in order to obtain new

Here, the main purpose is to apply this theorem to MP-matrices as a method to bind their eigenvalues, depending on the characteristics of the MP ODE system, and, even more, the

For instance, if we consider the MP corresponding to Figure 5, the first ODE of Eq. (17) can be

This equation—which obviously corresponds to an input tank—gives the first row of the MP-

Now, if a flux balance is performed in this input tank, we have this equation: Φ<sup>0</sup> þ Φ<sup>21</sup> þ Φ<sup>31</sup> ¼ Φ<sup>12</sup> þ Φ13, and then Φ<sup>21</sup> þ Φ<sup>31</sup> < Φ<sup>12</sup> þ Φ<sup>13</sup> (at least if we consider the nontrivial case Φ<sup>0</sup> > 0). As a consequence of this fact, j j a<sup>11</sup> > R1, and the Gershgorin disk will

Now, if the second ODE of Eq. (17) is considered, this equation can be written as

This internal tank equation corresponds to the second row of the MP-matrix (Eq. (18)):

<sup>V</sup><sup>1</sup> C<sup>1</sup>

V1

Φ<sup>21</sup> V<sup>1</sup>

.

Φ<sup>31</sup> V1

Square Matrices Associated to Mixing Problems ODE Systems

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

51

<sup>V</sup><sup>1</sup> < 0 with radius

<sup>V</sup><sup>1</sup> <sup>C</sup><sup>3</sup> � ð Þ <sup>Φ</sup>12þΦ<sup>13</sup>

The Gershgorin disk corresponding to this row is centered at <sup>a</sup><sup>11</sup> ¼ � ð Þ <sup>Φ</sup>12þΦ<sup>13</sup>

one of the closed disks , called Gershgorin disks.

results about matrices corresponding to chemical problems.

<sup>V</sup><sup>1</sup> <sup>C</sup><sup>2</sup> <sup>þ</sup> <sup>Φ</sup><sup>31</sup>

It is easy to observe that the outgoing flux Φ<sup>0</sup> only appears in the last entry of the MP-matrix and the incoming flux Φ<sup>0</sup> only is involved in the first entry of vector B. These facts—particularly the first one—are relevant when applying the Gershgorin circle theorem, which will be exposed in the next section.
