Stirred vessels, with inlet and outlet fluid streams, are widely used as chemical reactors in practice. Their behavior can be approximated by an ideal model: the continuous-flow stirred tank reactor (CSTR), also called the perfectly mixed flow reactor, where temperature and concentration are uniform in the entire vessel and equal to those of the outlet stream. This device is particularly suited for processes where temperature and composition should be controlled and a significant amount of heat of reaction removed. Examples include nitration of aromatic hydrocarbons or glycerin, production of ethylene glycol, copolymerization of butadiene and styrene, polymerization of ethylene using a Ziegler catalyst, hydrogenation of α-methylstyrene to cumene, and air oxidation of cumene to acetone and phenol (Froment and Bischoff, 1990).

Although CSTRs are simple devices, they can exhibit a parametrically sensitive behavior when exothermic reactions are carried out. In this case, conversion and temperature in the reactor undergo large variations in response to small variations of one or more of the reactor operating conditions. Therefore, in practice, we need to determine operating conditions that avoid the parametrically sensitive region.

The parametric sensitivity behavior of CSTRs has been investigated only recently. The main reason is that in this case there is neither a temperature profile nor a hot spot, as in the case of batch or tubular reactors, so that all earlier parametric sensitivity criteria based on some geometric feature of these temperature profiles cannot be applied. It is only through the calculation of sensitivity of model outputs with respect to model inputs that the sensitivity behavior of such lumped systems can be investigated.

The Concept of Sensitivity

The behavior of a chemical system is affected by many physicochemical parameters. Changing these parameters, we can alter the characteristics of the system to realize desired behavior or to avoid undesired behavior. In general, different parameters affect a system to different extents, and for the same parameter, its effect may depend on the range over which it is varied. By parametric sensitivity, we mean the sensitivity of the system behavior with respect to changes in parameters.

Let us illustrate the concept of sensitivity using some examples. Figure 1.1 shows the effect of changes in the initial temperature on the temperature evolution in a batch reactor for acetic anhydride hydrolysis, measured experimentally by Haldar and Rao (1992). There is a critical change in the temperature profile as the initial temperature increases from 319.0 to 319.5 K. In particular, an increase in the initial temperature by 0.5 K leads to a change in the temperature maximum by about 31 K. This experimental observation indicates that the system temperature becomes sensitive to small variations in the initial temperature in a specific region, called the parametrically sensitive region.

Figure 1.2 shows similar sensitivity phenomena in a tubular reactor obtained by numerical computations, given by Bilous and Amundson (1956) in their pioneering work on parametric sensitivity in the context of chemical reactors. In this example, the ambient temperature of a tubular reactor, where an exothermic reaction occurs, is changed. It is seen in Fig. 1.2a that when the ambient temperature increases by 2.5 K from 335 to 337.5 K, the temperature maximum (hot spot) along the reactor length changes by about 70 K.

Detailed or rigorous kinetic models, consisting of a large number of elementary reactions, are used increasingly to simulate complex reacting processes. An example is given in Chapter 7, where, using detailed kinetic models, we predicted the explosion limits of hydrogen-oxygen mixtures. The main advantage of a detailed versus a simplified or empirical kinetic model is its wider operating window. In other words, detailed models generally describe the kinetics of complex processes for a larger range of operating conditions, while simplified models can be used only for specific conditions. Moreover, detailed models are able to provide proper estimation of the radical concentrations involved in complex processes. Thus, detailed kinetic modeling is an important tool for the analysis and design of complex reacting systems.

A related aspect of detailed kinetic models is that, although they may provide satisfactory simulations of experimental results, their complexity often prevents the understanding of the key features of a process. For example, when using detailed kinetic models, it is often difficult to identify the main reaction paths in a complex reacting system.

Simplified kinetic models, on the other hand, offer several advantages in practical applications. A complex reacting (e.g., combustion) process typically involves a few hundreds of elementary reactions, and hence includes several hundred kinetic parameters. This is true not only for the combustion of complex fuels but also for simpler ones, such as hydrogen or methane. The computational effort associated with the application of complex kinetic models forces the introduction of very simple models to describe the transport processes in reactors where the reaction takes place (e.g., plug-flow or perfectly mixed).

Fixed-bed catalytic reactors consist of single tubes or bundles of tubes, packed with catalyst particles. They can be simulated by using the pseudo-homogeneous model where interparticle and intraparticle mass and heat transport resistances are neglected. The reactor runaway behavior predicted by this model has been discussed in Chapter 4. However, this simple model can be confidently applied to simulate catalytic reactors only for slow reactions. In most cases of practical interest, catalytic reactions are relatively fast and the roles of interparticle and intraparticle transport resistances have to be considered in the simulations. Thus, in the present chapter, the parametric sensitivity behavior of fixed-bed catalytic reactors is investigated by using a heterogeneous model, where both interparticle and intraparticle mass and heat transport resistances are included.

In a heterogeneous catalytic reactor, the temperature inside the catalyst particle is the key variable to be controlled, since it affects the reaction rate as well as the catalyst activity, selectivity, and life. Runaway of the particle temperature may occur because the fluid temperature is running away and the particle temperature simply follows it. This is the same phenomenon as in homogeneous tubular reactors discussed in Chapter 4. In this chapter we also account for the runaway of the particle temperature, which is strictly related to the heterogeneous nature of the system and is governed by the interaction between interparticle and intraparticle mass and heat transport and the chemical reactions.

Two approaches have been typically adopted to investigate the runaway behavior of the particle temperature. In the first, proposed by McGreavy and Adderley (1973), a single catalyst particle is extracted from the reactor and the temperature runaway behavior of this isolated particle is investigated.

Combustion processes are of central importance in a variety of applications, such as engines, turbines, and furnaces. The ignition of a combustion process can be either endogenous (i.e., self-ignition) or exogenous (i.e., induced by an external agent such as a spark or a local temperature increase). The identification of the self-ignition conditions for a given chemical system is not only of practical interest, but it is also a challenging test for the validation of combustion kinetic models. The first fundamental question that we address in this chapter is the definition of a criterion to establish whether a system has been ignited or not. As we will see in the following, this can be done by using the concepts related to parametric sensitivity discussed in previous chapters in the context of chemical reactors. As a model system we will use hydrogen oxidation, since it constitutes a prototype for more complex combustion processes, and its kinetic behavior has been well studied both experimentally and theoretically.

Ignition can be considered as a transition region or a boundary that separates slow from fast combustion processes. For combustion occurring in a shock tube or in a closed vessel, it is often required to determine the so-called ignition limits in a parameter space (typically temperature, pressure, and composition) that identify where the system is ignited, i.e., it undergoes a fast combustion process. For combustion induced by an external ignition source, a threshold needs to be defined to estimate the minimum energy required to ignite the system.

In the recent decades, growth in molecular biology has been explosive. The details of molecular constituents and of chemical transformations in metabolism have become increasingly clear, at least for a number of simpler organisms. An important strategy for molecular biologists is to reduce a complex biochemical reacting system to its elemental units, in order to explain it at the molecular level, and then to use this knowledge to reconstruct it. However, cellular components exhibit large interactions that are associative rather than additive. We still know relatively little about such interactions, what makes such an integrated system a living cell, and how a reconstructed system will respond to variations in novel environments and to specific variations in the metabolic pathway. To understand them, molecular biologists now recognize the need for systematic methods that can be provided by mathematical analysis.

As an effective mathematical tool, sensitivity analysis has been widely applied in biochemical reacting systems in order to understand the effects of variations in activities of enzymes, in kinetic parameters, and in external modifiers on metabolic processes. In other words, the objective is to understand how the function of an integrated biochemical system can be deduced from kinetic observations of its elemental units. The applications of sensitivity analysis in biochemical reacting systems were pioneered by Higgins (1963) and have spawned three basic theories: biochemical systems theory (Savageau, 1969a,b,c, 1971a,b, 1976), metabolic control theory (Kacser and Burns, 1973; Heinrich and Rapoport, 1974a,b; Westerhoff and Chen, 1984; Fell and Sauro, 1985; Reder, 1988; Giersch, 1988), and flux-oriented theory (Crabtree and Newsholme, 1978, 1985, 1987).

The prediction of pollutant distribution in the atmosphere from emission sources and meteorological fields is a primary objective of air pollution studies. In general, this requires solving the so-called Eulerian atmospheric species diffusion-reaction equations, which describe the time evolution of the pollutant concentration in the atmosphere in the presence of wind, diffusion processes, reaction, and source terms. These models tend to be complex, involving many physicochemical parameters and complex reaction mechanisms, whose quantitative evaluation is frequently not straightforward. Thus, in order to assess the reliability of a model, it is important to evaluate the influence of uncertainties in physicochemical, kinetic, and meteorological parameters on model predictions. This can be done conveniently using sensitivity analysis, which often accompanies the solution of atmospheric diffusion-reaction models. An important problem, particularly for air quality control, is to determine the influence of a specific source on a specific target location (usually referred to as a receptor). This type of information is not given directly by the solution of the Eulerian model, but can be obtained from sensitivity analysis.

The sensitivity analysis of model predictions with respect to uncertainties in input parameters has been described in previous chapters, where we examined local sensitivities that account for small one-at-a-time parameter variations. In order to investigate air pollution problems noted above, we first need to introduce some new concepts of sensitivity analysis. In particular, in Section 9.2, we illustrate a specific technique that is well suited to evaluate the relations between receptor and emission sources, where the latter may vary in space and time. This is still a type of local sensitivity analysis, but with the sensitivity definition based on functional rather than partial derivatives.

Some dictionaries define an engineer as a builder of engines, and it is relatively easy to classify engineering fields using this definition. Purists will insist that modern engineers rarely dirty their hands actually building anything; however, we can, without loss of the thread being developed here, include the design of engines within the definition. For example, many electrical engineers build (design) electrical engines such as motors and generators, and automotive engineers often build internal combustion engines. We can abstract the concept of engine to include machines in general as well as complex machines such as robots and vehicles. To further the abstraction, we can include systems that transfer or convert matter or energy from one state to another under the umbrella of machine design. Examples of such systems are water treatment facilities, the domain of civil engineers, or automated manufacturing facilities that attract the attention of industrial engineers. A computer is nothing more than an information processing engine. Now the material to be processed has been taken to the highest level of abstraction, the symbolic level.

The complexity of the world we live in, with the astonishingly high rate of information exchange and shrinking global barriers, demands that engineers utilize and command information processing systems. Computers are at the core of all nonbiological information processing systems, and they process the information that they are given with strict attention to detail. The level of detail is extreme, and the process by which we specify the details of the task that we wish the computer to perform is called programming.

Computer programming is the process by which we instruct a computer to perform a useful calculation or process. The computer can easily be described as an idiot savant, a term used by psychiatry to describe mental conditions in which individuals are capable of performing incredible feats of memory or calculation on request but are unable to understand the simplest activities of daily life. It is important to know that the computer will do only what it is instructed to do, no more and no less. Computers, sophisticated as they are, do not possess sentience or self-awareness. They cannot guess or anticipate what you want them to do; they simply do as instructed. There is a very old saying “garbage in, garbage out” that means if you put bad data into a program you can hardly expect to get good data out. Likewise, if your program is formed badly, the computer will not correct it for you.

Modern computers are rarely plagued by problems that cause them not to execute their instructions or to execute their instructions in a fashion other than that specified – in other words, they either work or they don't. If they seem to be acting strangely your program most likely is at fault. The most difficult task in the programming of a computer is the understanding of what the computer is supposed to do. To help make this task easier, we will examine techniques for describing problems and developing computer solutions.