Model complete mathematical

AVith this model the mathematical problems are greatly simplified since the radial variable is eliminated completely, decreasing the number... 

Where no complete mathematical description of the process and no dimensionless-numbers equations are available, modeling based on individual ratios can be employed. This is the most characteristic case for a number of industrial processes, especially in the field of organic-chemicals technology. This method is referred to as scale-up modeling (Mukhyonov et al., 1979). In such cases, individual ratios for the model and the object, which should have a constant value, are employed. For instance, there should be a constant ratio between the space velocity of the reacting mixture in the model and the industrial object. Some of the dimensionless numbers mentioned in physical modeling are also employed in this case. 

Beside the theoretically derived Gibbs adsorption isotherm, a large number of models have been developed that empirically describe a relationship between the interfacial coverage, the surface tension, and the surfactant concentration in the bulk phase. These adsorption isotherms are known under the names of the authors that first described them—i.e., the Fangmuir, Frumkin, or Volmer isotherms. A complete mathematical description of these isotherms is beyond the scope of this unit and the reader is encouraged to consult the appropriate literature instead (e.g., Dukhin et al., 1995). 

Then, the complete mathematical model can be stated as follows ... 

A quantitative kinetic model of the polymerization of a-pyrrolidine and cyclo(ethyl urea) showed,43 that two effects occur the existence of two stages in the initiation reaction and the absence of an induction period and self-acceleration in a-pyrrolidine polymerization. It was also apparent that to construct a satisfactory kinetic model of polymerization, it was necessary to introduce a proton exchange reaction and to take into consideration the ratio of direct and reverse reactions. As a result of these complications, a complete mathematical model appears to be rather difficult and the final relationships can be obtained only by computer methods. Therefore, in contrast to the kinetic equations for polymerization of e-caprolactam and o-dodecalactam discussed above, an expression... 

The importance of modeling batch processing systems forces a review of the mathematical analysis needed to set up and solve the models. The mathematical definition of physical problems involves (1) identification, (2) expression of the problem in mathematical language, (3) finding a solution, and (4) evaluating the solution. The completion of these steps in the order established determines whether a solution can be attained. The problem must be identified before one spends time setting up equations these and the initial and boundary conditions that define the problem must be well established before a solution is attempted then a solution can be obtained and evaluated. 

Aside from the continuity assumption and the discrete reality discussed above, deterministic models have been used to describe only those processes whose operation is fully understood. This implies a perfect understanding of all direct variables in the process and also, since every process is part of a larger universe, a complete comprehension of how all the other variables of the universe interact with the operation of the particular subprocess under study. Even if one were to find a real-world deterministic process, the number of interrelated variables and the number of unknown parameters are likely to be so large that the complete mathematical analysis would probably be so intractable that one might prefer to use a simpler stochastic representation. A small, simple stochastic model can often be substituted for a large, complex deterministic model since the need for the detailed causal mechanism of the latter is supplanted by the probabilistic variation of the former. In other words, one may deliberately introduce simplifications or errors in the equations to yield an analytically tractable stochastic model from which valid statistical inferences can be made, in principle, on the operation of the complex deterministic process. 

Computer simulation models have been formulated for cascade and Stratco sulfuric acid alkylation units. These complete models incorporate mathematical descriptions of all the interacting parts of the units, including reactors, distillation columns, compressors, condensers, and heat exchangers. Examples illus-strate diverse model applications. These Include identifying profitable unit modifications, comparing cascade to Stratco performance, evaluating optimal unit capacity and determining optimal deisobutanizer operation. 

In the fourth step of the building of a mathematical model of a process the assemblage of the parts (if any) is carried out in order to obtain the complete mathematical model of the process. Now the model dimension can be appreciated and a frontal analysis can be made in order to know whether analytical solutions are possible. 

Buckingham and Pople refer to the effect of the electric field as a paramagnetic term, and it has the dependence of the second term in equation (5), Although equation (5) has the virtue of attempting to describe the true electronic environment of the proton, it has the disadvantages of intractability. The electric field perturbation model is mathematically simple but an extreme approximation. Since these two treatments lead to the same functional dependence on p, perhaps the electric field model provides a useful approximation to the more complete description of equation (5), Whether this proves to be true or whether the characteristic arbitrariness of the electrostatic model will deprive the model of more than qualitative predictive value is not yet clear. In any event, the two treatments do concur in shifting attention from the p" term to the p term with its opposite sign. 

The reader will find here a complete mathematical development of the models of chromatography and other physical laws which direct the chemical engineer in the design and scale-up of chromatographic processes. For preparative chromatographic separations, our ultimate purpose is the optimization of the experimental conditions for maximum production rate, minimum solvent consumption, or minimum production cost, with or without constraints on the recovery yield. The considerable amormt of work done on this critical topic is presented in the... 

A complete mathematical model to describe a batch sohd-state fermentation for cellulase production in a fixed bed should include the following ... 

The equations presented above can be used (with or without modifications) to describe mass transfer processes in cocurrent flow. See, for example, the work of Modine (1963), whose wetted wall column experiments formed the basis for Example 11.5.3 and are the subject of further discussion in Section 15.4. The coolant energy balance is not needed to model an adiabatic wetted wall column and must be replaced by an energy balance for the liquid phase. Readers are asked to develop a complete mathematical model of a wetted wall column in Exercise 15.2.1. 

Before setting to solve this mathematical problem, we should note that while the model is mathematically sound and the question asked is meaningful, it cannot represent a complete physical system. If the Hamiltonian was a real representation of a physical system we could never prepare the system in state 11). Still, we shall see that this model represents a situation which is ubiquitous in molecular systems, not necessarily in condensed phase. Below we outline a few physical problems in which our model constitutes a key element ... 

The proposed methodology for computer-aided optimal design in the development of YSZ-based gas sensors comprises three phases. Firstly, the complete mathematical model with distributed temporal and spatial parameters for electrochemical gas sensors is presented as a system of the differential equations in private derivatives of parabolic and hyperbolic types. The complexity of physical and chemical interactions, represented in this model, allows performing a mathematical description of the electrochemical gas sensors toward standardization of the calculating procedures. The complete mathematical model and the algorithm of transfer from the complete... 

By using the deduction principle, the complete mathematical model of the electrochemical gas sensors with distributed parameters can be transformed to the mathematical models of the specific gas sensors, which is important for organization of their optimal design. 

Based on analysis of equations (2.1)-(2.4) of the complete mathematical model, it can be confirmed that some of the parameters of the YSZ-basedgas sensors (initial concentration of the charge carriers, concentration of the measuring gas, working temperature, etc.) are included directly into the system of equation (2.1). Other parameters, for example, thickness of the SE, are included into th boundary conditions (2.3) or (2.4). In this case, some of the functions (/i, /2, f/o (jf)), characterizing the interactions of the measuring environment with the SE (RE) of the sensor, must be taken into consideration at determination of the elementary and boundary conditions for the complete model (2.1)-(2.4). 

FIGURE 2.2 Algorithm of the transfer from the complete mathematical model with distributed parameters to the models of the real gas sensors. (Reprinted from Zhuiykov, S., Mathematical modelling of YSZ-based potentiometric gas sensors with oxide sensing electrodes part II Complete and numerical models for analysis of sensor characteristics. Sensors and Actuators B, Chem. 120 (2007) 645-656, with permission from Elsevier Science.)... 

Following 2005-2006 pubhcations, the comprehensive analysis of processes occurring at the TPB in the presence of water vapor for the YSZ-based NO2 sensor with the NiO-SE has been done by using the complete mathematical model... 

Zhuiykov, S., Complete mathematical model of electrochemical gas sensors with distributed temporal and spatial parameters, in Proc. Ilth Int. Meeting on Chemical Sensors, 11-17 July 2006, Brescia, Italy, 203. 

The most complete mathematical model of a nonuniform adsorbed layer is the distributed model, which takes into account interactions of adsorbed species, their mobility, and a possibility of phase transitions under the action of adsorbed species. The layer of adsorbed species corresponds to the two-dimensional model of the lattice gas, which is a characteristic model of statistical mechanics. Currently, it is widely used in the modeling of elementary processes on the catalyst surface. The energies of the lateral interaction between species localized in different lattice cells are the main parameters of the model. In the case of the chemisorption of simple species, each species occupies one unit cell. The catalytic process consists of a set of elementary steps of adsorption, desorption, and diffusion and an elementary act of reaction, which occurs on some set of cells (nodes) of the lattice. 

As with many model developments for fouling mechanisms the mathematical analysis is valuable since it draws attention to the salient effects of the variables. The present state-of-the-art models put forward, only go so far towards a complete mathematical solution that may be incorporated in the design of heat exchangers operating under chemical reaction fouling conditions. They are useful however, in suggesting a basis for an empirical formula to correlate experimental data that might so be used. 

The combination of the models described above provides a fairly complete mathematical description of the optical lithographic process. In the following sections, we discuss each of these models in depth, including their mathematical derivations, as well as the physical basis for their applications. ... 

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