The concept of stability is extremely important for a wide variety of dynamical systems, ranging from purely mathematical to natural and social systems. Physical and engineering systems, such as chemical reaction systems, are usually designed to operate at an equilibrium state (closed systems) or steady state (open systems), which must be stable for successful operation. This stability depends on whether a small perturbation is damped, leading to a stable state, or amplified, which results in an unstable state.

In chemical reactors, an enormous variety of possible regimes, both non-steady-state (transient) and steady-state can be observed. Steady-state reaction rates can be characterized by maxima and hystereses. Non-steady-state kinetic dependences may exhibit many phenomena of complex behavior, such as “fast” and “slow” domains, ignition and extinction, oscillations and chaotic behavior. These phenomena can be even more complex taking into account transport processes in three-dimensional media. In this case, waves and different spatial structures can be generated.

Diffusion has a history in science of more than one and a half centuries (Philibert, 2005, 2006). Driven by the experimental observation of the German chemist Johann Döbereiner in 1823 that hydrogen gas diffuses faster than air, in the late 1820s and early 1830s the Scottish chemist Thomas Graham performed a systematic study on diffusion in gases and found that the rate of gas diffusion is inversely proportional to the square root of its molecular mass (Sherwood et al., 1971).

Kinetic experiments are experiments in which the change of chemical composition of the reacting mixture is monitored in time and/or as a function of the various parameters of the system, in particular the inlet or initial chemical composition, temperature, catalyst state, etc. The main paradigm of contemporary chemical kinetics is the following: a chemical reaction is complex and consists of a set of elementary reactions for which the kinetic laws are assumed to be known.

In this chapter, the problem of optimization of multizone configurations is dealt with. When talking about a “zone,” we consider a thin active zone in which perfect mixing is assumed. Typically, thin catalyst zones are used in temporal analysis of products studies, but other types of experiments involving thin active zones are also conceivable. Thin active zones are separated by inactive transport zones. Here we consider transport to be purely diffusional, and the inactive zone to be purely diffusive.

Chemistry needs mathematics. Mathematics needs chemistry. Since the Dark Ages, in chemistry, or alchemy as it was called then, the so-called cross rule was well known for answering questions regarding the preparation of solutions of a certain desired concentration.

Given a set of components of known atomic composition, establish which of them are key to determine, together with the element balances, the amounts of all others. Furthermore, find a way of generating all possible reactions involving these components.

The Newton-Gauss method assumes that with good approximation f can be assumed to be linear over the full length of every parameter step. It is also possible, however, to search for the minimum of the objective function, Eq. (9.4), numerically without having to make such an assumption by using algorithms derived from optimization theory.

The principle of critical simplification in chemical kinetics is formulated using algebraic methods. Relaxation procedures for distinguishing linear and non-linear models are described. Intersections and coincidences in kinetic dependences are analyzed.

This chapter gives an overview of the most frequently applied numerical methods for the simulation of polymerization processes, that is, the calculation of the polymer microstructure as a function of monomer conversion and process conditions such as the temperature and initial concentrations. It is important to note, such simulations allow one to optimize the macroscopic polymer properties and to influence the polymer processability and final polymer product application range. Both deterministic and stochastic modeling techniques are discussed. In deterministic modeling techniques, time variation is seen as a continuous and predictable process, whereas in stochastic modeling techniques, a random-walk process is assumed instead.