Coupled Simultaneous ODE

In principle, any set of n linear first order coupled equations is equivalent to the Mth order inhomogeneous equation given earlier as 

These definitions turn the nth order equation into the coupled set of first order equations 

Treatment of elementary matrix methods is reviewed in Appendix A at the end of this book. Students interested in advanced material on the subject should consult the excellent text by Amundson (1966). Suffice to say that these specialized techniques must ultimately solve the same required characteristic equation as taught here, namely, 

However, useful methods exist that treat simultaneous equations without resorting to formalized methods of multilinear algebra. We shall discuss two of these methods because of their utility and frequent occurrence in practical problems  

These common-sense methods often escape the notice of an analyst, because the structure and complexity of a problem may be so intimidating. We illustrate the above principles with a few examples as follows. 

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Coupled Simultaneous ODE 93 operator D = d/dx. Thus, rewrite using operators to see... 

Equations (8.21) and (8.22) constitute a set of simultaneous ODEs in the independent variable z. The dependent variables are the a(i) terms. Each ODE is coupled to the adjacent ODEs i.e., the equation for a(i) contains a(i— 1) and a(i+ 1). Equation (8.24) is a special, degenerate member of the set, and Equation (8.22) for a(0) is also special because, due to symmetry, there is only one adjacent point, a(l). The overall set may be solved by any desired method. Euler s method is discussed below and is illustrated in Example 8.5. There are a great variety of commercial and freeware packages available for solving simultaneous ODEs. Most of them even work. Packages designed for stiff equations are best. The stiffness arises from the fact that V- i) becomes very small near the tube wall. There are also software packages that will handle the discretization automatically. 

If we have two simultaneous, coupled ODEs to numerically integrate... 

We see that we have j coupled ordinary differential equations that imist be solved simultaneously with either a numerical package or by writing an ODE solver. In fact, this procedure has been developed to take advantage of the vast number of computation techniques now available on mainframe (e.g,... 

When depends on a alone, the ODE is variable separable and can usually be solved analytically. If depends on the concentration of several components (e.g., a second-order reaction of the two-reactant variety, (Ra = —kab), versions of Equations 1.23 and 1.24 can be written for each component and the resulting equations solved simultaneously. Alternatively, stoichiometric relations can be used to couple the concentrations, but this approach becomes awkward in multicomponent systems and is avoided by the methodology introduced in Chapter 2. 

The PDF has been converted to a set of ODEs. Each version of Equation 8.30 governs the behavior of a as it evolves along a line of constant r. The independent variable is z. The main dependent variable is a(r) at location r. Equation 8.30 contains other dependent variables, a r + Ar) and a r — Ar), which are the main dependent variables along their own Tines. These side variables couple the set of equations so that they must be solved simultaneously. 

The first of these assumptions drops the momentum terms from the equations of motion, giving a situation known as creeping flow. This leaves and coupled through a pair of simultaneous PDEs. The pair can be solved when circumstances warrant, but the second assumption allows much greater simplification. It allows an uncoupling of the two equations so that is given by a single ODE ... 

This novel design shategy to simulate and control thermal runaway in a doublepipe reactor requires the simultaneous solution of four coupled first-order ODEs to describe conversion and temperature profiles within the inner pipe and in the annular region. Mass and thermal energy balances for exothermic reaction within the inner pipe are exactly the same as those discussed above (see equations 4-62 and 4-63). Hence, for one exothermic reaction (i.e., A products) in the inner pipe. 

With reference to item (4), quite often the ODEs generated by separation of variables do not produce easy analytical solutions. Under such conditions, it may be easier to solve the PDE by approximate or numerical methods, such as the orthogonal collocation technique, which is presented in Chapter 12. Also, the separation of variables technique does not easily cope with coupled PDE, or simultaneous equations in general. For such circumstances, transform methods have had great success, notably the Laplace transform. Other transform methods are possible, as we show in the present chapter. 

Simultaneous solution of the so-called differential-algebraic equation (DAE) set requires coupling of the ODE and algebraic equation solvers, the latter which are not discussed here, but can be found in detail elsewhere [ 1 ]. Description of a DAE set and its solution in the context of a one-dimensional (ID) heterogeneous packed-bed reactor model for autothermal conversion of methane to hydrogen is available in the literature [7]. It is also worth noting that packages such as DASSL and DAEPACK are also available for the solution of coupled DAE sets. 

This means that the number of equations in the kinetic systems of ODEs is equal to the number of species in the reaction mechanism. These equations are coupled and therefore can only be solved simultaneously. It is also generally true that in order to accurately represent the time-dependent behaviour of a chemical system, the ODEs should be based on the chemical mechanism incorporating intermediate species and elementary reaction steps rather than the overall reaction equation which contains only reactants and products. We will see later in Chap. 7 that one aim of chemical mechanism reduction is to limit the number of required intermediates within the mechanism in order to reduce the number of ODEs required to accurately represent the time-dependent behaviour of key species. 

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