Other Mathematical Models

Abemative approaches that have received attention include the tadform film and blind-side channel models. Lr the uniform film model [Kuo, 1960] b is asaimed that the initial displacement stage leaves a channel for wash liquor with a stagnant fihn of solutebearing liquid on the pore surfrce. The subsequent removal of solute is determined by mass trans across the stagnant film boundary. Various assunq)tions are made such as plug flow in the channel, no sorption by the particles and no axial dispersion. 

Mass solute balances on a differential slice of pore and the stagnant film give partial differential equations  

The solution to these equations for the solute concentration in the wash liquor is  

Conq aiison of these models with experimratal results has shown a match over part of the washing curve on some occadons however both are difficult to appfy and for these reasons are not preferred to the di erdon modd. 

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Empirical modek Empirical models rely on the correlation of atmospheric dispersion data for characteristic release types. Two examples of empirically based models are the Pasquill-Ginord model (for passive contaminants) and the Britter-McQuaid model (for denser-than-air contaminants) both of which are described below. Empirical models can be useful for the validation of other mathematical models but are limited to the characteristic release scenarios considered in the correlation. Selected empirical models are discussed in greater detail below because they can provide a reasonable first approximation of the hazard extent for many release scenarios and can be used as screening tools to indicate which release scenarios are most important to consider. 

If we consider the set of six equations on mass and momentum conservation on the one hand, and the characteristic energy extrema for stability of the three broad regimes of operation on the other, mathematical modeling for local hydrodynamics of particle-fluid two-phase flow beyond minimum fluidization needs therefore to satisfy the following constraints ... 

Relative recovery can be mathematically expressed by Fick s law of diffusion as modified by Jacobson (Jacobson et al., 1985). In this relationship, recovery is the ratio of the concentration in the perfusate to the concentration extracellular. For the mass recovery, an expression similar to a Michaelis-Menten formula for enzymatic reactions was derived (Ekblom et al., 1992). A number of other mathematical models for quantitative microdialysis have been proposed and are reviewed elsewhere (Justice 1993 Kehr, 1993b). 

In addition to the EMG fimction, many other mathematical models have been suggested to accoimt for the profiles of experimental peaks and to determine characteristic shape-parameters, such as a number of theoretical plates, a skew and an excess. These parameters are related to the second, third and fourth moments (Eq. 6.77) of the peak, respectively. For example, the Gram-Charlier series (GC) [96,114,115] and the Edgeworth-Cramer series (EC) [115,116] have been... 

The mentioned uniqueness of the Fade approximant for the given input Maclaurin series (4) represents a critical feature of this method. In other words, the ambiguities encountered in other mathematical modelings are eliminated from the outset already at the level of the definition of the PA. Moreover, this definition contains its "figure of merit" by revealing how well the PA can really describe the function G(z ) to be approximated. More precisely, given the infinite sum G(2 ) via Eq. (4), the key question to raise is about the best agreement between from Eq. (2) and G(z ) from... 

Other mathematical models of tolerance distributions which produce a sigmoid appearance of their corresponding dose-response functions have been suggested. The most commonly used is the log logistic function. 

In these cases, as in other mathematical models corresponding to ODE linear systems, stability and/or asymptotical stability strongly depend on the eigenvalues, particularly on their signs and multiplicity. 

Although the pioneering studies in QSPR-QSAR theory were established by Wiener in 1947 [1-4], other mathematical models had been reported previously for the prediction of the properties of substances. For example, it is well known that simple additive schemes and group contribution methods were used before the first QSPR-QSAR analyses [5-7]. We begin by summarizing these approaches, simply for the fact that optimal descriptors have gained some insight from these particular methods. 

As a graphical tool, Petri nets capture and represent the dynamic and concurrent properties of system. It is possible to set up state equations, algebraic equations, and other mathematical models governing the behaviour of systems. Petri nets can model a variety of situations and are easy to understand. However, the underlying model is difficult to solve. A larger model may become very eonqrlex and a solution may require the use of Monte Carlo tools. 

Mathematical models can be believed and even known, involving no evidential uncertainty. Consider e g. the Pythagorean theorem this is a mathematical piece of knowledge. Other mathematical models, e g. numerical solutions to differential equations, may be accepted rather than believed, and there may be evidential uncertainty e.g. related to the adopted discretization in the numerical solution. 

While physical modeling has been important historically, and is still a central part of chemical education and some investigations in stereochemistry, contemporary chemical models are almost always mathematical. Families of partially overlap>-ping, partially incompatible models such as the valence bond, molecular orbital, and semi-empirical models are used to explain and predict molecular structure and reactivity. Molecular mechanical models are used to explain some aspects of reaction kinetics and transport processes. And lattice models are use to explain thermodynamic properties such as phase. These and other mathematical models are ubiquitous in chemistry textbooks and articles, and chemists see them as central to chemical theory. 

When considering mathematical models of plates and shells, the authors clearly perceived the necessity for a reasonable compromise so that, on the one hand, the used models should describe the principle of a physical phenomenon and, on the other, they should be quite simple in order that the mathematical tool could be usefully employed. 

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. 

Transport Models. Many mechanistic and mathematical models have been proposed to describe reverse osmosis membranes. Some of these descriptions rely on relatively simple concepts others are far more complex and require sophisticated solution techniques. Models that adequately describe the performance of RO membranes are important to the design of RO processes. Models that predict separation characteristics also minimize the number of experiments that must be performed to describe a particular system. Excellent reviews of membrane transport models and mechanisms are available (9,14,25-29). 

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). 

Mathematically speaking, a process simulation model consists of a set of variables (stream flows, stream conditions and compositions, conditions of process equipment, etc) that can be equalities and inequalities. Simulation of steady-state processes assume that the values of all the variables are independent of time a mathematical model results in a set of algebraic equations. If, on the other hand, many of the variables were to be time dependent (m the case of simulation of batch processes, shutdowns and startups of plants, dynamic response to disturbances in a plant, etc), then the mathematical model would consist of a set of differential equations or a mixed set of differential and algebraic equations. 

Theoretically based correlations (or semitheoretical extensions of them), rooted in thermodynamics or other fundamentals are ordinarily preferred. However, rigorous theoretical understanding of real systems is far from complete, and purely empirical correlations typically have strict limits on apphcabihty. Many correlations result from curve-fitting the desired parameter to an appropriate independent variable. Some fitting exercises are rooted in theory, eg, Antoine s equation for vapor pressure others can be described as being semitheoretical. These distinctions usually do not refer to adherence to the observations of natural systems, but rather to the agreement in form to mathematical models of idealized systems. The advent of readily available computers has revolutionized the development and use of correlation techniques (see Chemometrics Computer technology Dimensional analysis). 

It may not be possible to develop a mathematical model for the fourth problem it not enough is known to characterize the performance of a rod versus the amounts of the various ingredients used in its manufacture. The rods may have to be manufactured and judged by ranking the rods relative to each other, perhaps based partially or totally on opinions. Pattern search methods have been devised to attack problems in this class. 

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... 

In maldug electrochemical impedance measurements, one vec tor is examined, using the others as the frame of reference. The voltage vector is divided by the current vec tor, as in Ohm s law. Electrochemical impedance measures the impedance of an electrochemical system and then mathematically models the response using simple circuit elements such as resistors, capacitors, and inductors. In some cases, the circuit elements are used to yield information about the kinetics of the corrosion process. 

Measurement Selection The identification of which measurements to make is an often overlooked aspect of plant-performance analysis. The end use of the data interpretation must be understood (i.e., the purpose for which the data, the parameters, or the resultant model will be used). For example, building a mathematical model of the process to explore other regions of operation is an end use. Another is to use the data to troubleshoot an operating problem. The level of data accuracy, the amount of data, and the sophistication of the interpretation depends upon the accuracy with which the result of the analysis needs to oe known. Daily measurements to a great extent and special plant measurements to a lesser extent are rarelv planned with the end use in mind. The result is typically too little data of too low accuracy or an inordinate amount with the resultant misuse in resources. 

For field-oriented controls, a mathematical model of the machine is developed in terms of rotating field to represent its operating parameters such as /V 4, 7, and 0 and all parameters that can inlluence the performance of the machine. The actual operating quantities arc then computed in terms of rotating field and corrected to the required level through open- or closed-loop control schemes to achieve very precise speed control. To make the model similar to that lor a d.c. machine, equation (6.2) is further resolved into two components, one direct axis and the other quadrature axis, as di.sciis.sed later. Now it is possible to monitor and vary these components individually, as with a d.c. machine. With this phasor control we can now achieve a high dynamic performance and accuracy of speed control in an a.c. machine, similar to a separately excited d.c. machine. A d.c. machine provides extremely accurate speed control due to the independent controls of its field and armature currents. 

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