Mathematical Modeling Studies

Chern [42] developed a mechanistic model based on diffusion-controlled reaction mechanisms to predict the kinetics of the semibatch emulsion polymerization of styrene. Reasonable agreement between the model predictions and experimental data available in the literature was achieved. Computer simulation results showed that the polymerization system approaches Smith-Ewart Case 2 kinetics (n = 0.5) when the concentration of monomer in the latex particles is close to the saturation value. By contrast, the polymerization system under the monomer-starved condition is characterized by the diffusion-con-trolled reaction mechanisms (n 0.5). The author also developed a model to predict the effect of desorption of free radicals out of the latex particles on the kinetics of the semibatch emulsion polymerization of methyl acrylate [43]. The validity of the kinetic model was confirmed by the experimental data for a wide range of monomer feed rates. The desorption rate constant for methyl acrylate at 50°C was determined to be 4 x 10 cm s  

Based on the coagulative particle nucleation mechanism, a two-step model was developed for the semibatch surfactant-free emulsion polymerization of 

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Fixed-bed reactors are used for testing commercial catalysts of larger particle sizes and to collect data for scale-up (validation of mathematical models, studying the influence of transport processes on overall reactor performance, etc.). Catalyst particles with a size ranging from 1 to 10 mm are tested using reactors of 20 to 100 mm ID. The reactor diameter can be decreased if the catalyst is diluted by fine inert particles the ratio of the reactor diameter to the size of catalyst particles then can be decreased to 3 1 (instead of the 10 to 20 recommended for fixed-bed catalytic reactors). This leads to a lower consumption of reactants. Very important for proper operation of fixed-bed reactors, both in cocurrent and countercurrent mode, is a uniform distribution of both phases over the entire cross-section of the reactor. If this is not the case, reactor performance will be significantly falsified by flow maldistribution. 

Trapaga and Szekely 515 conducted a mathematical modeling study of the isothermal impingement of liquid droplets in spray processes using a commercial CFD code called FLOW-3D. Their model is similar to that of Harlow and Shannon 397 except that viscosity and surface tension were included and wetting was simulated with a contact angle of 10°. In a subsequent study, 371 heat transfer and solidification phenomena were also addressed. These studies provided detailed... 

It seems to me that there are four types of studies available in the literature mathematical model studies, cost added studies, empirical studies and case studies. The distinction between each and their value to our subject is worth noting. 

Nattel S, Kneller J, Zou R, Leon LJ. Mechanisms of termination of atrial fibrillation by Class I antiarrhythmic drugs evidence from clinical, experimental, and mathematical modeling studies. J Cardiovasc Electro-physiol 2 003 14(suppl) S133—S139. 

Fitting the model to the observed data is an important task. Each mathematical model studied consists of independent and dependent variables, constants (possibly), and parameters that have to be estimated. The objective is to reduce the overall difference between the observed data and the calculated points by adjusting the values of the parameters. As mentioned earlier, the methods to be discussed assume that there is no error in the independent variable. Also, in general, the criteria of best fit will be the weighted sum of squared residuals between the observed and calculated data, the WSS. The chosen model can be validated in part by accurately describing the observed data. 

Whereas technical or inherent elements can be studied by experiments which allow, at least to some extent, reasonable extrapolations, it is difficult to scientifically evaluate the role and impact of management factors and also the value of product use strategies. Since the publication of the first mathematical model studies by Kable and Jeffery (181, several computer-based models for resistance management have been developed (9, 19, 20, 21, 22, 23. 

Real-life scheduling problems usually are very different from the mathematical models studied by researchers in academia and industrial research centers. It is difficult to categorize all differences between the real problems and the theoretical models, as each real-life scheduling problem has its own idiosyncrasies. Nevertheless, a number of these differences do stand out and are worth mentioning. 

E.g., in devefoping this theory we assume that some quantities may be arbitrary reals (cf. application of Lemma A.5.f from AppendixA.5 in Sect. 2.2) though we know that aU such possible values are far out of the Umits of practical apphcabihty of the mathematical model studied. 

Lactose Operom A Mathematical Modeling Study and Comparison with Experimental Data. 

The aims of the research project have been to obtain information on the rates and mechanisms for a range of VOC photo-oxidation reactions under simulated atmospheric conditions. By deriving detailed quantitative kinetic data on the reactions involved in the complex chemistry of the tropospheric photodegradations of VOC, we seek to provide key parameters required in tropospheric modelling. Such mathematical modelling studies will ultimately form the basis for drawing up scientifically sound control strategies for the release of VOC into the atmosphere. 

So the value of the outcome of the mathematical modeling study depends on ... 

Dias, C., Portela, M. and Bond, G. (1996). Oxidation of o-Xylene to Phthalic Anhydride over V2O5/T1O2 Catalysts. Part 4. Mathematical Modelling Study and Analysis of the Reaction Network, J. Catal, 164, pp. 276-287. 

More recently, Solsvik and Jakobsen [140] performed a numerical study comparing several closures for mass diffusion fluxes of multicomponent gas mixtures the Wilke, Maxwell-Stefan, dusty gas, and Wilke-Bosanquet models, on the level of the single catalyst pellet and the impacts of the mass diffusion flux closures employed for the pellet, on the reactor performance. For this investigation, the methanol synthesis operated in a fixed packed bed reactor was the chemical process adopted. In the mathematical modeling study of a novel combined catalyst/sorbent pellet. Rout et al. [121] investigated the performance of the sorption-enhanced steam methane reforming (SE-SMR) process at the level of a single pellet. Different closures... 

Experimental and mathematical modeling studies were performed to evaluate the potential benefits and limitations associated with the use of nonionic surfactants to enhance the microbial transformation of hexachlorobenzene (HCB) by a dechlorinating mixed culture enriched from a contaminated sediment. In general. Tween series surfactants were shown to have little impact on methanogenesis, whereas, polyoxyethylene (POE) alcohols, Triton X-100 and SDS were found to strongly inhibit methanogenesis and HCB dechlorination. Subsequent experiments conducted with Tween 80 illustrated the ability of this surfactant to enhance the solubility of HCB and to reduce the HCB-soil distribution coefficient. Model simulations demonstrated, however, that the aqueous phase mass fraction of HCB was substantially reduced in micellar solutions, which corresponded with observed reductions in HCB dechlorination. These results indicate that the impacts of surfactants on both biological activity and contaminant phase distributions should be evaluated in order to accurately assess the potential for biotransformation of hydrophobic contaminants in the presence of surfactants. 

Two other broad areas of food preservation have been studied with the objective of developing predictive models. En2yme inactivation by heat has been subjected to mathematical modeling in a manner similar to microbial inactivation. Chemical deterioration mechanisms have been studied to allow the prediction of shelf life, particularly the shelf life of foods susceptible to nonen2ymatic browning and Hpid oxidation. 

Solubihties of 1,3-butadiene and many other organic compounds in water have been extensively studied to gauge the impact of discharge of these materials into aquatic systems. Estimates have been advanced by using the UNIFAC derived method (19,20). Similarly, a mathematical model has been developed to calculate the vapor—Hquid equiUbrium (VLE) for 1,3-butadiene in the presence of steam (21). 

At times, it is possible to build an empirical mathematical model of a process in the form of equations involving all the key variables that enter into the optimisation problem. Such an empirical model may be made from operating plant data or from the case study results of a simulator, in which case the resultant model would be a model of a model. Practically all of the optimisation techniques described can then be appHed to this empirical model. 

The scope of a study required to satisfy these goals will be dependent upon the extent of the risk, the depth of the study required, and the level of resources available (mathematical models and tools and skilled people to perform the study and any internal or external constraints). 

The accuracy of absolute risk results depends on (1) whether all the significant contributors to risk have been analyzed, (2) the realism of the mathematical models used to predict failure characteristics and accident phenomena, and (3) the statistical uncertainty associated with the various input data. The achievable accuracy of absolute risk results is very dependent on the type of hazard being analyzed. In studies where the dominant risk contributors can be calibrated with ample historical data (e.g., the risk of an engine failure causing an airplane crash), the uncertainty can be reduced to a few percent. However, many authors of published studies and other expert practitioners have recognized that uncertainties can be greater than 1 to 2 orders of magnitude in studies whose major contributors are rare, catastrophic events. 

Empirical energy functions can fulfill the demands required by computational studies of biochemical and biophysical systems. The mathematical equations in empirical energy functions include relatively simple terms to describe the physical interactions that dictate the structure and dynamic properties of biological molecules. In addition, empirical force fields use atomistic models, in which atoms are the smallest particles in the system rather than the electrons and nuclei used in quantum mechanics. These two simplifications allow for the computational speed required to perform the required number of energy calculations on biomolecules in their environments to be attained, and, more important, via the use of properly optimized parameters in the mathematical models the required chemical accuracy can be achieved. The use of empirical energy functions was initially applied to small organic molecules, where it was referred to as molecular mechanics [4], and more recently to biological systems [2,3]. 

The potential energy function presented in Eqs. (2) and (3) represents the minimal mathematical model that can be used for computational studies of biological systems. Currently,... 

Central to the quality of any computational smdy is the mathematical model used to relate the structure of a system to its energy. General details of the empirical force fields used in the study of biologically relevant molecules are covered in Chapter 2, and only particular information relevant to nucleic acids is discussed in this chapter. 

The reader is encouraged to use a two-phase, one spatial dimension, and time-dependent mathematical model to study this phenomenon. The UCKRON test problem can be used for general introduction before the particular model for the system of interest is investigated. The success of the simulation will depend strongly on the quality of physical parameters and estimated transfer coefficients for the system. 

Catalytic crackings operations have been simulated by mathematical models, with the aid of computers. The computer programs are the end result of a very extensive research effort in pilot and bench scale units. Many sets of calculations are carried out to optimize design of new units, operation of existing plants, choice of feedstocks, and other variables subject to control. A background knowledge of the correlations used in the "black box" helps to make such studies more effective. 

Coker, A. K., Mathematical Modelling and a Study of Flow Patterns in Cylindrical Nozzle, M.Sc. Tliesis, Aston University, 1979. 

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 ... 

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-... 

Chapters 15 through 17 are devoted to mathematical modeling of particular systems, namely colloidal suspensions, fluids in contact with semi-permeable membranes, and electrical double layers. Finally, Chapter 18 summarizes recent studies on crystal growth process. 

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