Mathematical models reactions

For all reactions, the mass transport regime is controlled by the diffusion of the reacting ligand only, as the mercury electrode serves as an inexhaustible source for mercury ions. Hence, with respect to the mathematical modeling, reactions (2.205) and (2.206) are identical. This also holds true for reactions (2.210) and (2.211). Furthermore, it is assumed that the electrode surface is covered by a sub-monomolecular film without interactions between the deposited particles. For reactions (2.207) and (2.209) the ligand adsorption obeys a linear adsorption isotherm. Assuming semi-infinite diffusion at a planar electrode, the general mathematical model is defined as follows ... 

Experimental data of stearic acid decarboxylation in a laboratory-scale fixed bed reactor for formation of heptadecane were evaluated studied with the aid of mathematical modeling. Reaction kinetics, catalyst deactivation, and axial dispersion were the central elements of the model. The effect of internal mass transfer resistance in catalyst pores was found negligible due to the slow reaction rates. The model was used for an extensive sensitivity study and parameter estimation. With optimized parameters, the model was able to describe the experimentally observed trends adequately. A reactor scale-up study was made by selecting the reactor geometry (diameter and length of the reactor, size and the shape of the catalyst particles) and operating conditions (superficial liquid velocity, temperature, and pressure) in such a way that nonideal flow and mass and heat transfer phenomena in pilot scale were avoided. 

An R-matrix has a series of interesting matheinatical properties that directly reflect chemical laws. Thus, the sum of all the entries in an R-matrix must be zero, as no electrons can be generated or annihilated in a chemical reaction. Furthermore, the sum of the entries in each row or column of an R-matrix must also he zero as long as there is not a change in formal charges on the corresponding atom. An elaborate mathematical model of the constitutional aspects of organic chemistry has been built on the basis of BE- and R-matriccs [17. ... 

The production of acetic acid from butane is a complex process. Nonetheless, sufficient information on product sequences and rates has been obtained to permit development of a mathematical model of the system. The relationships of the intermediates throw significant light on LPO mechanisms in general (22). Surprisingly, ca 25% of the carbon in the consumed butane is converted to ethanol in the first reaction step. Most of the ethanol is consumed by subsequent reaction. 

Those based on strictly empirical descriptions Mathematical models based on physical and chemical laws (e.g., mass and energy balances, thermodynamics, chemical reaction kinefics) are frequently employed in optimization apphcations. These models are conceptually attractive because a gener model for any system size can be developed before the system is constructed. On the other hand, an empirical model can be devised that simply correlates input-output data without any physiochemical analysis of the process. For... 

The reacting sohd is in granular form. Decrease in the area of the reaction interface occurs as the reaction proceeds. The mathematical modeling is distinguished from that with flat surfaces, which are most often used in experimentation. 

One mathematical model of the oxidation of nickel spheres was confirmed when it took into account the decrease in the reaction surface as the reaction proceeded. 

In this work the development of mathematical model is done assuming simplifications of physico-chemical model of peroxide oxidation of the model system with the chemiluminesce intensity as the analytical signal. The mathematical model allows to describe basic stages of chemiluminescence process in vitro, namely spontaneous luminescence, slow and fast flashes due to initiating by chemical substances e.g. Fe +ions, chemiluminescent reaction at different stages of chain reactions evolution. 

The mathematical model was based on the scheme utilized in chemiluminescent method that was supplement with the reactions of radicals, formed of inhibitor molecules - AO. 

A numerical value is obtainable by integrating the trend curve for the flow received from the Flow Recorder (FR), from the start of the reaction to a time selected. Doing this from zero to each of 20 equally spaced times gives the conversion of the solid soda. Correlating the rates with the calculated X s, a mathematical model for the dependence of rate on X can be developed. 

To facilitate the use of methanol synthesis in examples, the UCKRON and VEKRON test problems (Berty et al 1989, Arva and Szeifert 1989) will be applied. In the development of the test problem, methanol synthesis served as an example. The physical properties, thermodynamic conditions, technology and average rate of reaction were taken from the literature of methanol synthesis. For the kinetics, however, an artificial mechanism was created that had a known and rigorous mathematical solution. It was fundamentally important to create a fixed basis of comparison with various approximate mathematical models for kinetics. These were derived by simulated experiments from the test problems with added random error. See Appendix A and B, Berty et al, 1989. 

In a continuous reaction process, the true residence time of the reaction partners in the reactor plays a major role. It is governed by the residence time distribution characteristic of the reactor, which gives information on backmixing (macromixing) of the throughput. The principal objectives of studies into the macrokinetics of a process are to estimate the coefficients of a mathematical model of the process and to validate the model for adequacy. For this purpose, a pilot plant should provide the following ... 

In this chapter The background of shock-induced solid-state ehemistry eonceptual models and mathematical models chemical reactions in shock-compressed porous powders sample preservation. 

Horie and his coworkers [90K01] have developed a simplified mathematical model that is useful for study of the heterogeneous nature of powder mixtures. The model considers a heterogeneous mixture of voids, inert species, and reactant species in pressure equilibrium, but not in thermal equilibrium. The concept of the Horie VIR model is shown in Fig. 6.3. As shown in the figure, the temperatures in the inert and reactive species are permitted to be different and heat flow can occur from the reactive (usually hot) species to the inert species. When chemical reaction occurs the inert species acts to ther-... 

There exist many different CA models exhibiting BZ-like spatial waves. One of the simplest, and earliest, described in the next section, is a model proposed by Greenberg and Hastings in 1978 [green78], and based on an earlier excitable media model by Weiner and Rosenbluth [weiner46]. One of the earliest and simplest mathematical models of the BZ reaction, called the Orcgonator, is due to Field and Noyes [field74]. 

As our first approach to the model, we considered the controlling step to be the mass transfer from gas to liquid, the mass transfer from liquid to catalyst, or the catalytic surface reaction step. The other steps were eliminated since convective transport with small catalyst particles and high local mixing should offer virtually no resistance to the overall reaction scheme. Mathematical models were constructed for each of these three steps. 

The principal difficulty with these equations arises from the nonlinear term cb. Because of the exponential dependence of cb on temperature, these equations can be solved only by numerical methods. Nachbar has circumvented this difficulty by assuming very fast gas-phase reactions, and has thus obtained preliminary solutions to the mathematical model. He has also examined the implications of the two-temperature approach. Upon careful examination of the equations, he has shown that the model predicts that the slabs having the slowest regression rate will protrude above the material having the faster decomposition rate. The resulting surface then becomes one of alternate hills and valleys. The depth of each valley is then determined by the rate of the fast pyrolysis reaction relative to the slower reaction. 

Gal-Or and Hoelscher (G5) have recently proposed a mathematical model that takes into account interaction between bubbles (or drops) in a swarm as well as the effect of bubble-size distribution. The analysis is presented for unsteady-state mass transfer with and without chemical reaction, and for steady-state diffusion to a family of moving bubbles. 

C.G. Vayenas, S. Brosda, and C. Pliangos, Rules and Mathematical Modeling of Electrochemical and Chemical Promotion 1. Reaction Classification and Promotional Rules,/. Catal., in press (2001). 

The parameter a in Equation (11.6) is positive for electrophobic reactions (5r/5O>0, A>1) and negative for electrophilic ones (3r/0Oelectrochemical promotion behaviour is frequently encountered, leading to volcano-type or inverted volcano-type behaviour. However, even then equation (11.6) is satisfied over relatively wide (0.2-0.3 eV) AO regions, so we limit the present analysis to this type of promotional kinetics. It should be remembered thatEq. (11.6), originally found as an experimental observation, can be rationalized by rigorous mathematical models which account explicitly for the electrostatic dipole interactions between the adsorbates and the backspillover-formed effective double layer, as discussed in Chapter 6. 

A mathematical model for this polymerization reaction based on homogeneous, isothermal reaction is inadequate to predict all of these effects, particularly the breadth of the MWD. For this reason a model taking explicit account of the phase separation has been formulated and is currently under investigation. 

Mathematical models of the reaction system were developed which enabled prediction of the molecular weight distribution (MWD). Direct and indirect methods were used, but only distributions obtained from moments are described here. Due to the stiffness of the model equations an improved numerical integrator was developed, in order to solve the equations in a reasonable time scale. 

Mathematical models of the reaction system have been developed, enabling prediction of the molecular weight... 

In some cases, the reaction may take place before the substrate is reached while still in the gas phase (gas-phase precipitation) as will be reviewed later. As can be expected, the mathematical modeling of these phenomena can be complicated. 

Operability analysis and control system synthesis for an entire chemical plant Mathematical modeling of transport and chemical reactions of combustion-generated air pollutants... 

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