Kinetic system of ODEs

A possible way to achieve this is to solve equation (4.1), together with all the sensitivity equations belonging to different parameters. In this case, the stiff ode solver has to decompose a large (m + l)n x (m + l)n Jacobian at each time-step, which is very inefficient. If the kinetic ode (4.1) is coupled with a single sensitivity equation at one time, the joint Jacobian is smaller, but the kinetic system of odes has to be solved m times. 

An alternative method is based on the investigation of the Jacobian of the kinetic system of odes, J = df/dc. A species may be considered redundant if its concentration change has no significant effect on the rate of production of important species. An element of the normed Jacobian (3 ln/,)/(3 In Cy) shows the fractional change of the rate of production of species i caused by the fractional change of the concentration of species j. The influence of the change of the concentration of species i on the rate of production of an A-membered group of important species can be taken into account by the sum of squares of normalized Jacobian elements,... 

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. 

In adiabatic systems or in systems with a known heat loss rate, usually temperature is added as the (As + l)fh variable of the kinetic system of ODEs. The differential equation for the rate of change of temperature in a closed spatially homogeneous reaction vessel is given as... 

The kinetic system of ODEs and its initial values together provide the following initial value problem ... 

From a mathematical point of view, the kinetic system of ODEs is first-order and usually nonlinear, since it contains first-order derivatives with respect to time and the time derivative is usually a nonlinear function of concentrations. In general. 

In theory, if a laboratory experiment is repeated say one hour later than the first execution, then the same concentration-time curves should be obtained (ignoring experimental error for now). Accordingly, the time in the kinetic system of differential equations is not the wall-clock time, but a relative time from the beginning of the experiment. Such a differential equation system is called an autonomous system of ODEs. In other cases, such as in atmospheric chemical or biological circadian rhythm models, the actual physical time is important, because a part of the parameters (the rate coefficients belonging to the photochemical reactions) depend on the strength of sunshine, which is a function of the absolute time. In this case, the kinetic system of ODEs is nonautonomous. 

The first example for the creation of the kinetic system of ODEs will be based on a skeleton hydrogen combustion mechanism. Using the law of mass action, the rates ri to rs of the reaction steps can be calculated from the species concentrations and... 

Let us determine the matrices J and F belonging to the kinetic system of ODEs above. These two types of matrices will be used several dozen times in the following chapters. For example, the Jacobian is used within the solution of stiff differential equations (Sect. 6.7), the calculation of local sensitivities (Sect. 5.2) and in timescale analysis (Sect. 6.2), whilst matrix F is used for the calculation of local sensitivities (Sect. 5.2). Carrying out the appropriate derivations, the following matrices are obtained ... 

Simplification of a kinetic mechanism or the kinetic system of ODES is often required in order to facilitate finding solutions to the resulting equations and can sometimes be achieved based on kinetic simplification principles. In most cases, the solutions obtained are not exactly identical to those from the fuU system of equations, but it is usually satisfactory for a chemical modeller if the accuracy of the simulation is better than the accuracy of the measurements. For example, usually better than 1 % simulation error for the concentrations of the species of interest when compared to the original model is appropriate. Historically, simplifications were necessary before the advent of computational methods in order to facilitate the analytical solution of the ODEs resulting from chemical schemes. We begin here by discussing these early simplification principles. In later chapters, we will introduce more complex methods for chemical kinetic model reduction that may perhaps require the application of computational methods. 

The quasi-steady-state approximation (QSSA) is also called the Bodenstein principle, after one of its first users (Bodenstein 1913). As a first step, species are selected that will be called quasi-steady-state (or QSS) species. The QSS-species are usually highly reactive and low-concentration intermediates, like radicals. The production rates of these species are set to zero in the kinetic system of ODEs. The corresponding right-hand sides form a system of algebraic equations. These... 

As noted above, the consideration of conserved properties allows the kinetic system of ODEs to contain fewer variables than the number of species. However, it is an exact transformation, and therefore it is usually handled separately from the rules above which are based on approximations. 

In general, the local sensitivity matrix can only be determined numerically. If the original system of kinetic differential equations can be solved numerically, then the local sensitivity matrix can also be calculated using finite-difference approximations (see Eq. (5.2)). To calculate the sensitivity matrix in this way, we have to know the original solution and the m solutions obtained by perturbing each parameter one by one. All in all, the kinetic system of ODEs has to be solved m + 1)... 

Although the PCAS and PCAF methods are similar in form, these two methods are fundamentally different. The objective function of PCAF contains the production rates of species, and the matrix F can be calculated from the right-hand side of ODE (2.9). The objective function of PCAS contains the concentrations of species [the solution of ODE (2.9)], and the matrix S has to be obtained from the solution of the sensitivity differential equations (5.7) and is therefore computationally more time consuming. Put another way, PCAS investigates the effect of parameter changes on the solution of the kinetic system of ODEs, whilst PCAF examines the effect of parameter changes on the right-hand sides of the kinetic system of ODEs (2.9). 

The dimension h of the new variable vector Y is smaller than that of the original concentration vector h kinetic system of ODEs is formed ... 

At the beginning of reaction kinetic simulations, usually the concentrations of only a few species (e.g. reactants, diluent gases, etc.) are defined, and other concentrations are set to zero. The QSSA is not usually applicable from the beginning of the simulation since at this point, the trajectories are quite far from any underlying slow manifolds (see Sect. 6.5). Hence, the kinetic system of ODEs (7.69) is usually solved first, and at time ti is switched to the solution of the DAE system (7.70-7.71). We denote Y(fi)=(Y Vi)> Y (fi)) to be the solution of Eq. (7.69) at time t. When the system of Eqs. (7.70-7.71) is used, then the concentrations of the QSS-species are calculated first via the solution of algebraic system of Eqs. (7.71), and the result is concentration vector The concentrations of... 

Fig. 7.11 Simulation of a skeletal model of the Belousov-Zhabotinsky reaction based on the solution of the kinetic system of ODEs (solid line) and using a repro-model (dots), (a) Concentration-time curves (b) the solution in phase space. Reprinted from Turanyi (1994) with permission from Elsevier...

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