ON THE POSSIBILITY OF DIRECT MODELING OF CHEMICAL-TECHNOLOGY PROCESSES ON ANALOG COMPUTERS USING DIRECTED GRAPHS

G. V. RADCHENKO, G. A. INESHIN, V. G. KHAN

(Presented by the scientific-technical seminar of the Department of Automation and Telemechanics)

Direct modeling proves possible in all cases in which it is possible to establish an isomorphism between the variables of a system and the vertices of a topologically directed graph [3]. In this work, conditions are shown for the existence of a graph for various systems. Scalar variables of two types, in addition to the properties given in the premises for representing technological processes by a graph, must also determine the dynamic properties of any part of the system if the parametric quantities are known. Moreover, it is not necessary that the systems under investigation be linear, since directed graphs are used here only as analog models, i.e., they may contain branches whose transfer coefficients are nonlinear or vary with time.

In the present work an attempt is made to apply the propositions given in [2, 3] to the modeling of chemical-technology processes. At the first stage, as a simple element of a technological scheme, an exchanger was chosen (a heat and mass exchanger). In an exchanger that is a four-terminal network (two input streams and two output streams), two variables are assigned to each stream. Since the equations expressing the basic laws of transfer (of heat and matter) are identical to the equation for the transfer of the quantity of electricity and to one another, the directed graphs in all three cases must be identical:

Fig. 1

\[ \partial Q=\chi \frac{dT}{dl}\, dS\, dt; \qquad \partial m=D\frac{dC}{dl}\, dS\, dt; \qquad dQ=\sigma \frac{dV}{dl}\, dS\, dt; \]

\[ Q=WpI \ (W=\varkappa St); \qquad M=\varkappa pC \ (\varkappa=DSt); \qquad Q=BpV \ (B=\sigma St). \]

Here the variables \(Q\) and \(I\)—the quantity of heat and enthalpy—are put in correspondence with the heat flux; the variables \(M\) and \(C\)—the quantity of matter and concentration—with the mass flux; and the variables \(Q\) and \(V\)—the quantity of electricity and potential—with the electric flux; \(p=\frac{d}{dt}\) is the differentiation operator in the direction of transfer.

Taking the foregoing into account, let us write the equations of mass and heat exchange (material and heat balance without allowance for losses). A conventional representation of an exchanger (the case of a heat exchanger) is shown in Fig. 1. In the representation in Fig. 1 it is assumed that

\[ I_1 = I + I_2;\quad Q_1 = Q + Q_2;\quad I_4 = I' + I_3; \]

\[ Q_4 = Q + Q_3;\quad Q = W_1 p I;\quad I' = \frac{1}{W_2 p}\, Q. \]

Fig. 2

Here \(I\) and \(Q\) are the enthalpy and the amount of heat transferred from the heated stream to the less heated one. The equations for mass exchange will be analogous:

\[ C_1 = C + C_2;\quad C_4 = C + C_3;\quad M_1 = M + M_2;\quad M_4 = M + M_3;\quad M = \kappa p C. \]

Here \(C\) and \(M\) are the content of the distributed component and the amount of substance transferred from the phase in which there is more of this component to the phase in which there is less of it. The graphs for heat and mass exchangers are shown in Fig. 2, \(a\) and 2, \(b\), respectively.

The elementary graphs of the four-terminal subnetworks are chosen so that they correspond to the solving blocks; therefore, the directed graph can be regarded as a circuit for assembling the model. Since the directed graph can be obtained from the equivalent circuit, one may speak of direct modeling of the physical system without preliminary compilation of the equations of the problem under investigation.

Fig. 3

To avoid the use of differentiators in the model, graphs of other types can be obtained for the same four-terminal networks by means of inversion.

Fig. 4.

Inscriptions in the figure:

• Still
• Stripping section
• Rectifying section
• Reflux condenser
• $t_{\mathrm{res}}$
• $Q_{\mathrm{res}}$
• $t_{\mathrm{dist}}$
• $Q_{\mathrm{dist}}$

For direct modeling of more complex objects, which cannot be represented solely by exchangers, one should construct a structural diagram consisting of elementary links, without violating the condition of isomorphism between the variables of the system and the vertices of the graph.

As an example, let us consider obtaining a model of the heat exchange of a rectification column. Figure 3 shows the substitution diagram. Replacing each element of the physical system (the substitution diagram) by its corresponding graph, we obtain a directed graph of the heat-exchange process (Fig. 4).

From the graph of the initial physical circuit, composed of elementary subgraphs, a direct transition is made to the model assembled from computing blocks, since the graphs of these blocks [3] are identical to the elementary subgraphs. Thus, the complete graph of the initial circuit is essentially the initial diagram of the analog model (Fig. 5).

Fig. 5

Conclusions

The consideration of the possibility of direct modeling of chemical-technology processes on analog computers by means of directed graphs leads to the conclusion that this method proves to be very convenient and may serve as a subject for further research.

References

1. M. Korakh, L. Gashko. Representation of chemical technology processes on the basis of graph theory. Theoretical Foundations of Chemical Technology, vol. 11, no. 3, 1968.

2. N. M. Trent. Isomorphisms between oriented linear graphs and limped physical system. Journal Acoust. Soc. Amer., 27, pp. 500—527, 1955.

3. L. Robisho, M. Buaver, J. Rober. Directed graphs and their applications to electrical circuits and machines. “Energiya,” 1964.

4. O. Ore. Graph Theory. Moscow, “Nauka,” 1968.

5. K. Berge. Graph Theory and Its Applications. Moscow, IL, 1962.