3. Previous results revisited

The system-associated matrix (MP-matrix) is

�ð Þ Φ<sup>12</sup> þ Φ<sup>13</sup> þ Φ<sup>14</sup> =V<sup>1</sup> 0 0 00

Hereafter, we will call MP-matrix to any ODE system-associated matrix related to a given MP,

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

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

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

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).

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

possible enumeration for this purpose is the one illustrated in Figure 1.

mixing problem without recirculation (MP-WR).

Otherwise, it will be an open system (MP-OS).

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

Φ12=V<sup>2</sup> �Φ25=V<sup>2</sup> 0 00 Φ13=V<sup>3</sup> 0 �Φ35=V<sup>3</sup> 0 0 Φ14=V<sup>4</sup> 0 0 �Φ45=V<sup>4</sup> 0

0 Φ25=V<sup>5</sup> Φ35=V<sup>5</sup> Φ45=V<sup>5</sup> �Φ0=V<sup>5</sup>

1

CCCCCCCCCA

(6)

A ¼

like matrix A of Eq. (6).

flux, i.e., it is a closed system.

are systems with recirculation.

0

44 Matrix Theory-Applications and Theorems

BBBBBBBBB@

In order to give some general results, it is convenient to consider two different situations: MP without recirculation and MP with recirculation.

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 compartment will be of the form:

$$\mathcal{W}\_{i}\frac{dC\_{j}}{dt} \quad \alpha\_{i1}C\_{i1} + \alpha\_{i2}C\_{i2} + ... + \alpha\_{ik}C\_{ik} \tag{7}$$

being and

As a consequence, under the previous conditions, the corresponding ODE system has an associated upper matrix.

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

$$\begin{cases} \frac{d\mathbf{C}\_1}{dt} = \frac{\Phi\_0}{V\_1}\mathbf{C}\_0 - \frac{(\Phi\_{12} + \Phi\_{13} + \Phi\_{14})}{V\_1}\mathbf{C}\_1\\ \frac{d\mathbf{C}\_2}{dt} = \frac{\Phi\_{12}}{V\_2}\mathbf{C}\_1 - \frac{\Phi\_{25}}{V\_2}\mathbf{C}\_2\\ \frac{d\mathbf{C}\_3}{dt} = \frac{\Phi\_{13}}{V\_3}\mathbf{C}\_1 - \frac{\Phi\_{35}}{V\_3}\mathbf{C}\_3\\ \frac{d\mathbf{C}\_4}{dt} = \frac{\Phi\_{14}}{V\_4}\mathbf{C}\_1 - \frac{\Phi\_{45}}{V\_4}\mathbf{C}\_4\\ \frac{d\mathbf{C}\_5}{dt} = \frac{\Phi\_{25}}{V\_5}\mathbf{C}\_2 + \frac{\Phi\_{35}}{V\_5}\mathbf{C}\_3 + \frac{\Phi\_{45}}{V\_5}\mathbf{C}\_4 - \frac{\Phi\_0}{V\_5}\mathbf{C}\_5 \end{cases} \tag{8}$$

It follows that for the jth compartment, the coefficient corresponding to can be written as

, where represents the sum of outgoing fluxes. This situation can be easily

generalized, since concentration only appears in the right hand side of the corresponding ODE when a certain flux is leaving the tank. Combining this result with the previous one about the upper matrix—it is easy to observe that the ODE system has only negative eigen-

$$\text{values of the form } \mathcal{A}\_j = \frac{-\sum\_{k} \Phi\_{jk}}{V\_j} < 0 \text{ for all } j = 1, 2, \dots, n$$

However, not all of these results can be extended to MPs with recirculation as will be analyzed in the following subsection.

In previous works [4, 5], a "black box" system was analyzed (see Figure 3), in order to obtain a necessary condition to be satisfied by any MP-matrix with any number of compartments and unknown internal geometry. In Figure 3 and represent flux and concentration at the input, and is also the output flux (since tanks neither fill up nor empty with time), and is the final concentration. In this system there are compartments inside the black box with volumes and concentrations , and recirculation fluxes may exist or not.

If all volumes remain constant, by performing a mass balance, it can be proved that.

$$\sum\_{i=1}^{n} V\_i \frac{dC\_i}{dt} = \Phi\_0 \left( C\_0 - C\_n \right) \tag{9}$$

dC<sup>1</sup>

8 >><

>>:

Operating with these equations, it can be proved that V<sup>1</sup>

by Eq. (9) is satisfied.

dition Eq. (9) since is zero.

It is easy to observe that

and A the MP-matrix.

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

A ¼

0

BB@

This equation can be written as <sup>V</sup><sup>T</sup> <sup>A</sup> <sup>¼</sup> <sup>0</sup>, being <sup>V</sup> the volumes' vector.

ð Þ V<sup>1</sup> V<sup>2</sup>

matrix corresponding to Figure 1, the result will be

dt ¼ � <sup>Φ</sup><sup>12</sup> V<sup>1</sup> C<sup>1</sup> þ Φ<sup>21</sup> V1 C2

> dC<sup>1</sup> dt þ V<sup>2</sup>

dC<sup>2</sup>

Square Matrices Associated to Mixing Problems ODE Systems

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CCA <sup>¼</sup> ð Þ 0 0 (12)

dt ¼ 0, which satisfies con-

(10)

47

(11)

<sup>C</sup><sup>1</sup> � <sup>Φ</sup><sup>21</sup> V2 C2

The previous result can be generalized as follows: in a given MP—with or without recirculation—with input and output concentrations and , respectively, and being the incoming and outgoing flux, then, independently of the internal geometry, the condition given

An analogous condition may be used to know if a given matrix may or may not be an MP-matrix. For this purpose, let us consider the MP-matrix , associated to the ODE system given by Eq. (10):

> � <sup>Φ</sup><sup>12</sup> V1

0

BB@

� <sup>Φ</sup><sup>12</sup> V1

> Φ<sup>12</sup> V2

Φ<sup>12</sup> V2

Φ<sup>21</sup> V1

1

CCA

<sup>V</sup><sup>T</sup> <sup>A</sup> <sup>¼</sup> ð Þ <sup>0000</sup> �Φ<sup>0</sup> (13)

<sup>V</sup><sup>T</sup> <sup>A</sup> <sup>¼</sup> ð Þ <sup>0</sup> <sup>⋯</sup> <sup>0</sup> �Φ<sup>0</sup> (14)

� <sup>Φ</sup><sup>21</sup> V2

1

Φ<sup>21</sup> V1

� <sup>Φ</sup><sup>21</sup> V2

If there exists an incoming (and outgoing) flux , the last result will change. For instance, if we compute <sup>V</sup><sup>T</sup> <sup>A</sup>, being <sup>V</sup> <sup>¼</sup> ð Þ <sup>V</sup><sup>1</sup> <sup>V</sup><sup>2</sup> <sup>V</sup><sup>3</sup> <sup>V</sup><sup>4</sup> <sup>V</sup><sup>5</sup> the volumes' vector and <sup>A</sup> the MP-

It can be noted that Eq. (12) and Eq. (13) are particular cases of the following result: in a given MP—with or without recirculation—with an incoming and outgoing flux , the condition <sup>V</sup><sup>T</sup> <sup>A</sup> <sup>¼</sup> ð Þ <sup>0</sup> <sup>⋯</sup> <sup>0</sup> �Φ<sup>0</sup> is satisfied, being <sup>V</sup> <sup>¼</sup> ð Þ <sup>V</sup><sup>1</sup> <sup>V</sup><sup>2</sup> <sup>⋯</sup> Vn the volumes' vector

Then, independently of the internal geometry of the system, the following condition is satisfied:

Then, Eq. (9) is obtained without any consideration of the internal geometry of the tank system and can be easily verified in the previous example (see Figure 1). In fact, by adding the equations of the ODE system (Eq. (4)), it follows straightforward that the condition given in Eq. (9) is satisfied. The same conclusion can be drawn from other possible examples, corresponding to open or closed MPs, with or without recirculation. For instance, in the case schematized in Figure 2, the ODE system can be written as follows:

Figure 3. A "black box" tank system.

Square Matrices Associated to Mixing Problems ODE Systems http://dx.doi.org/10.5772/intechopen.74437 47

$$\begin{cases} \frac{d\mathbb{C}\_1}{dt} = -\frac{\Phi\_{12}}{V\_1}\mathbb{C}\_1 + \frac{\Phi\_{21}}{V\_1}\mathbb{C}\_2\\ \frac{d\mathbb{C}\_2}{dt} = \frac{\Phi\_{12}}{V\_2}\mathbb{C}\_1 - \frac{\Phi\_{21}}{V\_2}\mathbb{C}\_2 \end{cases} \tag{10}$$

Operating with these equations, it can be proved that V<sup>1</sup> dC<sup>1</sup> dt þ V<sup>2</sup> dC<sup>2</sup> dt ¼ 0, which satisfies condition Eq. (9) since is zero.

The previous result can be generalized as follows: in a given MP—with or without recirculation—with input and output concentrations and , respectively, and being the incoming and outgoing flux, then, independently of the internal geometry, the condition given

$$\text{by Eq. (9)} \sum\_{i=1}^{n} V\_i \frac{dC\_i}{dt} = \Phi\_0 \left( C\_0 - C\_n \right) \text{is satisfied.}$$

An analogous condition may be used to know if a given matrix may or may not be an MP-matrix. For this purpose, let us consider the MP-matrix , associated to the ODE system given by Eq. (10):

$$\mathbf{A} = \begin{pmatrix} -\frac{\Phi\_{12}}{V\_1} & \frac{\Phi\_{21}}{V\_1} \\ \frac{\Phi\_{12}}{V\_2} & -\frac{\Phi\_{21}}{V\_2} \end{pmatrix} \tag{11}$$

It is easy to observe that

ð9Þ

It follows that for the jth compartment, the coefficient corresponding to can be written as

generalized, since concentration only appears in the right hand side of the corresponding ODE when a certain flux is leaving the tank. Combining this result with the previous one about the upper matrix—it is easy to observe that the ODE system has only negative eigen-

However, not all of these results can be extended to MPs with recirculation as will be analyzed

In previous works [4, 5], a "black box" system was analyzed (see Figure 3), in order to obtain a necessary condition to be satisfied by any MP-matrix with any number of compartments and unknown internal geometry. In Figure 3 and represent flux and concentration at the input, and is also the output flux (since tanks neither fill up nor empty with time), and is the final concentration. In this system there are compartments inside the black box with

volumes and concentrations , and recirculation fluxes may exist or not.

schematized in Figure 2, the ODE system can be written as follows:

If all volumes remain constant, by performing a mass balance, it can be proved that.

Then, Eq. (9) is obtained without any consideration of the internal geometry of the tank system and can be easily verified in the previous example (see Figure 1). In fact, by adding the equations of the ODE system (Eq. (4)), it follows straightforward that the condition given in Eq. (9) is satisfied. The same conclusion can be drawn from other possible examples, corresponding to open or closed MPs, with or without recirculation. For instance, in the case

values of the form , for all

in the following subsection.

46 Matrix Theory-Applications and Theorems

Figure 3. A "black box" tank system.

, where represents the sum of outgoing fluxes. This situation can be easily

$$
\begin{pmatrix}
(V\_1 \ V\_2) \begin{pmatrix} -\frac{\Phi\_{12}}{V\_1} & \frac{\Phi\_{21}}{V\_1} \\ \frac{\Phi\_{12}}{V\_2} & -\frac{\Phi\_{21}}{V\_2} \end{pmatrix} = \begin{pmatrix} 0 & 0 \end{pmatrix} \tag{12}
$$

This equation can be written as <sup>V</sup><sup>T</sup> <sup>A</sup> <sup>¼</sup> <sup>0</sup>, being <sup>V</sup> the volumes' vector.

If there exists an incoming (and outgoing) flux , the last result will change. For instance, if we compute <sup>V</sup><sup>T</sup> <sup>A</sup>, being <sup>V</sup> <sup>¼</sup> ð Þ <sup>V</sup><sup>1</sup> <sup>V</sup><sup>2</sup> <sup>V</sup><sup>3</sup> <sup>V</sup><sup>4</sup> <sup>V</sup><sup>5</sup> the volumes' vector and <sup>A</sup> the MPmatrix corresponding to Figure 1, the result will be

$$\mathbf{V}^{\mathsf{T}}\mathbf{A} = \begin{pmatrix} 0 & 0 & 0 & 0 & -\boldsymbol{\Phi}\_{0} \end{pmatrix} \tag{13}$$

It can be noted that Eq. (12) and Eq. (13) are particular cases of the following result: in a given MP—with or without recirculation—with an incoming and outgoing flux , the condition <sup>V</sup><sup>T</sup> <sup>A</sup> <sup>¼</sup> ð Þ <sup>0</sup> <sup>⋯</sup> <sup>0</sup> �Φ<sup>0</sup> is satisfied, being <sup>V</sup> <sup>¼</sup> ð Þ <sup>V</sup><sup>1</sup> <sup>V</sup><sup>2</sup> <sup>⋯</sup> Vn the volumes' vector and A the MP-matrix.

Then, independently of the internal geometry of the system, the following condition is satisfied:

$$\mathbf{V}^{\mathsf{T}}\mathbf{A} = \begin{pmatrix} 0 & \cdots & 0 & -\Phi\_0 \end{pmatrix} \tag{14}$$

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

$$\mathbf{A} = \begin{pmatrix} -\frac{\Phi\_{12}}{V\_1} & \frac{\Phi\_{21}}{V\_1} \\ \frac{\Phi\_{12}}{V\_2} & -\frac{\Phi\_{21}}{V\_2} \end{pmatrix} \tag{15}$$

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

$$\mathbf{A}\_{\ell} = \begin{pmatrix} -\frac{\Phi\_{12}}{V\_1} + \varepsilon & \frac{\Phi\_{21}}{V\_1} \\ \frac{\Phi\_{12}}{V\_2} & -\frac{\Phi\_{21}}{V\_2} \end{pmatrix} \tag{16}$$

the system and it also has an outgoing flux Φ<sup>0</sup> that leaves the tank system. It should be noted

Square Matrices Associated to Mixing Problems ODE Systems

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49

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

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

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

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

ð17Þ

that in Figure 4, the second tank is an internal one.

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

ones, and there is no internal tank.

among them, as in Figure 5.

can be formulated as follows:

for this problem.

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 independently of the internal geometry of the system.

As a first consequence, not every square matrix is an MP-matrix. A second observation is that if a given MP-matrix is slightly changed, the result is not necessarily a new MP-matrix.

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 exactly the same mathematical model.

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 answers.

The same situation can be observed in many other inverse problems [2], and it is not an exclusive property of compartment analysis.
