Mathematical model dynamic behavior

A mathematical model may be constructed representing a chemical reaction. Solutions of the mathematical model must be compatible with the observed behavior of this chemical reaction. Furthermore if some other solutions would indicate possible behaviors so far unobserved, of the reaction, experiments maybe designed to experimentally observe them, thus to reinforce the validity of the mathematical model. Dynamical systems such as reactions are modelled by differential equations. The chemical equilibrium states are the stable singular solutions of the mathematical model consisting of a set of differential equations. Depending on the format of these equations solutions vary in a number of possible ways. In addition to these stable singular solutions periodic solutions also appear. Although there are various kinds of oscillatory behavior observed in reactions, these periodic solutions correspond to only some of these oscillations. 

For the more mathematically inclined Investigate the dynamic behavior of a coupled linear three reservoir model using the technique outlined in Section 4.3.1. 

Atmospheric aerosols have a direct impact on earth s radiation balance, fog formation and cloud physics, and visibility degradation as well as human health effect[l]. Both natural and anthropogenic sources contribute to the formation of ambient aerosol, which are composed mostly of sulfates, nitrates and ammoniums in either pure or mixed forms[2]. These inorganic salt aerosols are hygroscopic by nature and exhibit the properties of deliquescence and efflorescence in humid air. That is, relative humidity(RH) history and chemical composition determine whether atmospheric aerosols are liquid or solid. Aerosol physical state affects climate and environmental phenomena such as radiative transfer, visibility, and heterogeneous chemistry. Here we present a mathematical model that considers the relative humidity history and chemical composition dependence of deliquescence and efflorescence for describing the dynamic and transport behavior of ambient aerosols[3]. 

The studies described in the preceding two sections have identified several processes that affect the dynamic behavior of three-way catalysts. Further studies are required to identify all of the chemical and physical processes that influence the behavior of these catalysts under cycled air-fuel ratio conditions. The approaches used in future studies should include (1) direct measurement of dynamic responses, (2) mathematical analysis of experimental data, and (3) formulation and validation of mathematical models of dynamic converter operation. 

The dynamic relationships discussed thus far in this book were determined from mathematical models of the process. Mathematical equations, based on fundamental physical and chemical laws, were developed to describe the time-dependent behavior of the system. We assumed that the values of all parameters, such as holdups, reaction rates, heat transfer coeflicients, etc., were known. Thus the dynamic behavior was predicted on essentially a theoretical basis. 

A much more interesting case of chaotic dynamics of the reactor can be obtained from the study of the self-oscillating behavior. Consider the simplified mathematical model (8) and suppose that the reactor is in steady state with a reactant concentration of Prom Eq.(8) the equilibrium point [x, y ] can be deduced as follows ... 

One approach to developing mathematical models is to begin with one that contains a relatively detailed description of the physical system and then to derive simpler models by identifying those elements that can be approximated while still retaining the essential behavior of the system (see, for example, Aris, 1978). This is the approach that we will follow here. Our particular interest will be in deriving mathematical models of packed bed reactors that are appropriate for use in designing control systems. Thus, we will be interested in models capable of simulating dynamic behavior. 

Tired of so much chemistry, some readers may now turn with pleasure to Part IV, in which transport and mixing phenomena are explained. Furthermore, Part IV provides the conceptual and mathematical framework for building models for the quantitative description of the dynamic behavior of organic chemicals in environmental systems. 

Ulrich, M. M., S. R. Mttller, H. P. Singer, D. M. Imboden, and R. P. Schwarzenbach, Input and dynamic behavior of the organic pollutants tetrachloroethene, atrazine, and NTA in a lake A study combining mathematical modeling and field measurements , Environ. Sci. Technol., 28, 1674-1685 (1994). 

In most adsorption processes the adsorbent is contacted with fluid in a packed bed. An understanding of the dynamic behavior of such systems is therefore needed for rational process design and optimization. What is required is a mathematical model which allows the effluent concentration to be predicted for any defined change in the feed concentration or flow rate to the bed. The flow pattern can generally be represented adequately by the axial dispersed plug-flow model, according to which a mass balance for an element of the column yields, for the basic differential equation governing llie dynamic behavior,... 

The single relaxation time approximation corresponds to a stochastic model in which the fluctuating force on a molecule has a Lorentzian spectrum. Thus if the fluctuating force is a Gaussian-Markov process, it follows that the memory function must have this simple form.64 Of course it would be naive to assume that this exponential memory will accurately account for the dynamical behavior on liquids. It should be regarded as a simple model which has certain qualitative features that we expect real memory functions to have. It decays to zero and, moreover, is of a sufficiently simple mathematical form that the velocity autocorrelation function,... 

In this fermentation process, sustained oscillations have been reported frequently in experimental fermentors and several mathematical models have been proposed. Our approach in this section shows the rich static and dynamic bifurcation behavior of fermentation systems by solving and analyzing the corresponding nonlinear mathematical models. The results of this section show that these oscillations can be complex leading to chaotic behavior and that the periodic and chaotic attractors of the system can be exploited for increasing the yield and productivity of ethanol. The readers are advised to investigate the system further. 

Mathematical modeling of the dynamic behavior of VOCs during showering indicated that the level of VOC exposure risk is determined to a large extent by the type of the showerhead used. Thus jet-flow showerheads were found to cause less exposure than spray type showerheads (Chen, Wu and Chang, 2003). 

The computer simulation is one of the essential means to investigate dynamic and steady-state behavior as well as control of metabolic pathways. A metabolic simulator is a computer program that performs one or several of the tasks including solving the steady state of a metabolic pathway, dynamically simulating a metabolic pathway, or calculating the control coefficient of a metabolic pathway. Its mathematical model generally consists of a set of differential equations derived from rate equations of the enzymatic reactions of the pathway. 

Mathematical models of biological processes are often used for hypothesis testing and process optimization. Using physical interpretation of results to obtain greater insight into process behavior is only possible when structured models that consider several parts of the system separately are employed. A number of dynamic mathematical models for cell growth and metabolite pro-... 

From a mathematical point of view, we can see that Equation (5.10) is in a (nonstandard) singularly perturbed form. This suggests that the integrated processes under consideration will feature a dynamic behavior with at least two distinct time scales. Drawing on the developments in Chapters 2, 3, and 4, the following section demonstrates that these systems evolve in effect over three distinct time scales and proposes a method for deriving reduced-order, non-stiff models for the dynamics in each time scale. 

This second-level modeling of the feedback mechanisms leads to nonlinear models for processes, which, under some experimental conditions, may exhibit chaotic behavior. The previous equation is termed bilinear because of the presence of the b [y (/,)] r (I,) term and it is the general formalism for models in biology, ecology, industrial applications, and socioeconomic processes [601]. Bilinear mathematical models are useful to real-world dynamic behavior because of their variable structure. It has been shown that processes described by bilinear models are generally more controllable and offer better performance in control than linear systems. We emphasize that the unstable inherent character of chaotic systems fits exactly within the complete controllability principle discussed for bilinear mathematical models [601] additive control may be used to steer the system to new equilibrium points, and multiplicative control, either to stabilize a chaotic behavior or to enlarge the attainable space. Then, bilinear systems are of extreme importance in the design and use of optimal control for chaotic behaviors. We can now understand the butterfly effect, i.e., the extreme sensitivity of chaotic systems to tiny perturbations described in Chapter 3. 

Nonsteady behavior of electrochemical systems was observed by -> Fechner as early as 1828 [ii]. Periodic or chaotic changes of electrode potential under - gal-vanostatic or open-circuit conditions and similar variation of -> current under potentiostatic conditions have been the subject of numerous studies [iii, iv]. The electrochemical systems, for which interesting dynamic behavior has been reported include anodic or open-circuit dissolution of metals [v-vii], electrooxidation of small organic molecules [viii-xiv] or hydrogen, reduction of anions [xv, xvi] etc. [ii]. Much effort regarding the theoretical description and mathematical modeling of these complex phenomena has been made [xvii-xix]. Especially studies that used combined techniques, such as radiotracer (- tracer methods)(Fig. 1) [x], electrochemi-... 

A useful literature relating to polypeptide and protein adsorption kinetics and equilibrium behavior in finite bath systems for both affinity and ion-ex-change HPLC sorbents is now available160,169,171-174,228,234 319 323 402"405 and various mathematical models have been developed, incorporating data on the adsorption behavior of proteins in a finite bath.8,160 167-169 171-174 400 403-405 406 One such model, the so-called combined-batch adsorption model (BAMcomb), initially developed for nonporous particles, takes into account the dynamic adsorption behavior of polypeptides and proteins in a finite bath. Due to the absence of pore diffusion, analytical solutions for nonporous HPLC sorbents can be readily developed using this model and its two simplified cases, and the effects of both surface interaction and film mass transfer can be independently addressed. Based on this knowledge, extension of the BAMcomb approach to porous sorbents in bath systems, and subsequently to packed-, expanded-, and fluidized-bed systems, can then be achieved. 

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