Differential equation graph model

Chapter 2 describes the evolution in fundamental concepts of chemical kinetics (in particular, that of heterogeneous catalysis) and the "prehis-tory of the problem, i.e. the period before the construction of the formal kinetics apparatus. Data are presented concerning the ideal adsorbed layer model and the Horiuti-Temkin theory of steady-state reactions. In what follows (Chapter 3), an apparatus for the modern formal kinetics is represented. This is based on the qualitative theory of differential equations, linear algebra and graphs theory. Closed and open systems are discussed separately (as a rule, only for isothermal cases). We will draw the reader s attention to the two results of considerable importance. 

In practice, a gray-box model is developed in steps. One early step is to decide which variables and interactions to include. This is often done by the sketching of an interaction-graph. It must then be decided if a variable should be a state or a dependent variable, and how the interactions should be formulated. In the case of metabolic reactions, the expression forms for the reactions have often been characterized in in-vitro experiments. If this has been done, there are also often in-vitro estimates of the kinetic parameters. For enzymatic networks, however, such in-vitro studies are much more rare, and it is hence typically less known which expression to choose for the reaction rates, and what a good estimate for the kinetic parameters is. In any case, the standard method of combining reaction rates, r,-, and an interaction graph into a set of differential equations is to use the stoichiometric coefficients, Sij... 

Given the rules, incidence matrix of the graph, and parameters defined above, we may derive the differential equations for the model directly using only matrix operations. 

Theorem 1 The system of differential equation for the model with the graph and parameters defined above is given by... 

While STELLA gives a glimpse of what future software might be able to do, modem spreadsheets are almost as easily used to solve comparable problems. Spreadsheets that can be linked to graphical output provide an extremely powerful environment for exploring numerical solutions to differential equations. Spreadsheets explicitly show the calculations, are easily set up and modified, and provide quick, dynamic graphs that make it easy to explore the effect of parameters on a model. 

Equation (4.51) can be represented by a linear /-field in derivative causality and modulated effort sinks accounting for the parameter variations added to 1-junctions. The incremental bond graph model of a linear 2-port /-field is depicted in Fig. 4.16. The prime denotes differentiation with respect to time. Accordingly, Fig. 4.17 shows the incremental bond graph of the circuit in Fig. 4.14. 

Keywords Automated modeling Simulation Mechatronics systems Computer generated differential equations Transfer functions State space CAMPG Bond graph Block diagrams MATLAB SIMULINK SYSQUAKE... 

The bond graph differential equations appear on a set of side-by-side individual windows which form a kind of system model control panel. On the left-hand side we have icons that link to subsystems representing each differential equation and each output variable. If one clicks on one of these icons representing the subsystems, the block diagram representing that differential equation or that output equation is displayed in a separate window. The system simulation control panel and two windows corresponding to two differential equations (dp2 and dq4) and two windows corresponding to two outputs (f3 and e5) are shown in Fig. 11.56. 

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