Mathematical modeling principles

The classical microscopic description of molecular processes leads to a mathematical model in terms of Hamiltonian differential equations. In principle, the discretization of such systems permits a simulation of the dynamics. However, as will be worked out below in Section 2, both forward and backward numerical analysis restrict such simulations to only short time spans and to comparatively small discretization steps. Fortunately, most questions of chemical relevance just require the computation of averages of physical observables, of stable conformations or of conformational changes. The computation of averages is usually performed on a statistical physics basis. In the subsequent Section 3 we advocate a new computational approach on the basis of the mathematical theory of dynamical systems we directly solve a... 

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. 

One of the major uses of molecular simulation is to provide useful theoretical interpretation of experimental data. Before the advent of simulation this had to be done by directly comparing experiment with analytical (mathematical) models. The analytical approach has the advantage of simplicity, in that the models are derived from first principles with only a few, if any, adjustable parameters. However, the chemical complexity of biological systems often precludes the direct application of meaningful analytical models or leads to the situation where more than one model can be invoked to explain the same experimental data. 

Theoretical representation of the behaviour of a hydrocyclone requires adequate analysis of three distinct physical phenomenon taking place in these devices, viz. the understanding of fluid flow, its interactions with the dispersed solid phase and the quantification of shear induced attrition of crystals. Simplified analytical solutions to conservation of mass and momentum equations derived from the Navier-Stokes equation can be used to quantify fluid flow in the hydrocyclone. For dilute slurries, once bulk flow has been quantified in terms of spatial components of velocity, crystal motion can then be traced by balancing forces on the crystals themselves to map out their trajectories. The trajectories for different sizes can then be used to develop a separation efficiency curve, which quantifies performance of the vessel (Bloor and Ingham, 1987). In principle, population balances can be included for crystal attrition in the above description for developing a thorough mathematical model. 

Mathews and Rawlings (1998) successfully applied model-based control using solids hold-up and liquid density measurements to control the filtrability of a photochemical product. Togkalidou etal. (2001) report results of a factorial design approach to investigate relative effects of operating conditions on the filtration resistance of slurry produced in a semi-continuous batch crystallizer using various empirical chemometric methods. This method is proposed as an alternative approach to the development of first principle mathematical models of crystallization for application to non-ideal crystals shapes such as needles found in many pharmaceutical crystals. 

Wave mechanics is based on the fundamental principle that electrons behave as waves (e.g., they can be diffracted) and that consequently a wave equation can be written for them, in the same sense that light waves, soimd waves, and so on, can be described by wave equations. The equation that serves as a mathematical model for electrons is known as the Schrodinger equation, which for a one-electron system is... 

A first principle mathematical model of the extruder barrel and temperature control system was developed using time dependent partial differential equations in cylindrical coordinates in two spatial dimensions (r and z). There was no angular dependence in the temperature function (3T/30=O). The equation for this model is (from standard texts, i.e. 1-2) ... 

Mathematical Models. As noted previously, a mathematical model must be fitted to the predicted results shown In each factorial table generated by each scientist. Ideally, each scientist selects and fits an appropriate model based upon theoretical constraints and physical principles. In some cases, however, appropriate models are unknown to the scientists. This Is likely to occur for experiments Involving multifactor, multidisciplinary systems. When this occurs, various standard models have been used to describe the predicted results shown In the factorial tables. For example, for effects associated with lognormal distributions a multiplicative model has been found useful. As a default model, the team statistician can fit a polynomial model using standard least square techniques. Although of limited use for Interpolation or extrapolation, a polynomial model can serve to Identify certain problems Involving the relationships among the factors as Implied by the values shown In the factorial tables. 

There are two basic classes of mathematical models (see Fig. 5.3-18) (1) purely empirical models, and (2) models based on physicochemical principles. 

Figure 5.3-18. Mathematical models according to their physical principles.

The consequence of all these (conscious and unconscious) simplifications and eliminations might be that some information not present in the process will be included in the model. Conversely, some phenomena occurring in reality are not accounted for in the model. The adjustable parameters in such simplified models will compensate for inadequacy of the model and will not be the true physical coefficients. Accordingly, the usefulness of the model will be limited and risk at scale-up will not be completely eliminated. In general, in mathematical modelling of chemical processes two principles should always be kept in mind. The first was formulated by G.E.P. Box of Wisconsin All models are wrong, some of them are useful . As far as the choice of the best of wrong models is concerned, words of S.M. Wheeler of New York are worthwhile to keep in mind The best model is the simplest one that works . This is usually the model that fits the experimental data well in the statistical sense and contains the smallest number of parameters. The problem at scale-up, however, is that we do not know which of the models works in a full-scale unit until a plant is on stream. 

Kapur (1988) has listed thirty-six characteristics or principles of mathematical modelling. These are very much a matter of common sense, but it is very important to have them restated, as it is often very easy to lose sight of the principles during the active involvement of modelling. They can be summarised as follows ... 

Principles of mathematical modelling 2 Probability density function 112 Process control examples 505-524 Product inhibition 643, 649 Production rate in mass balance 27 Profit function 108 Proportional... 

The experimentalist often formulates a mathematical model in order to describe the observed behavior. In general, the model consists of a set of equations based on the principles of chemistry, physics, thermodynamics, kinetics and transport phenomena and attempts to predict the variables, y, that are being measured. In general, the measured variables y are a function of x. Thus, the model has the following form... 

This closure property is also inherent to a set of differential equations for arbitrary sequences Uk in macromolecules of linear copolymers as well as for analogous fragments in branched polymers. Hence, in principle, the kinetic method enables the determination of statistical characteristics of the chemical structure of noncyclic polymers, provided the Flory principle holds for all the chemical reactions involved in their synthesis. It is essential here that the Flory principle is meant not in its original version but in the extended one [2]. Hence under mathematical modeling the employment of the kinetic models of macro-molecular reactions where the violation of ideality is connected only with the short-range effects will not create new fundamental problems as compared with ideal models. 

As this chapter aims at explaining the basics, operational principles, advantages and pitfalls of vibrational spectroscopic sensors, some topics have been simplified or omitted altogether, especially when involving abstract theoretical or complex mathematical models. The same applies to methods having no direct impact on sensor applications. For a deeper introduction into theory, instrumentation and related experimental methods, comprehensive surveys can be found in any good textbook on vibrational spectroscopy or instrumental analytical chemistry1"4. 

Molecular mechanics force fields rest on four fundamental principles. The first principle is derived from the Bom-Oppenheimer approximation. Electrons have much lower mass than nuclei and move at much greater velocity. The velocity is sufficiently different that the nuclei can be considered stationary on a relative scale. In effect, the electronic and nuclear motions are uncoupled, and they can be treated separately. Unlike quantum mechanics, which is involved in determining the probability of electron distribution, molecular mechanics focuses instead on the location of the nuclei. Based on both theory and experiment, a set of equations are used to account for the electronic-nuclear attraction, nuclear-nuclear repulsion, and covalent bonding. Electrons are not directly taken into account, but they are considered indirectly or implicitly through the use of potential energy equations. This approach creates a mathematical model of molecular structures which is intuitively clear and readily available for fast computations. The set of equations and constants is defined as the force... 

Software sensors and related methods - This last group is considered because of the complexity of wastewater composition and of treatment process control. As all relevant parameters are not directly measurable, as will be seen hereafter, the use of more or less complex mathematical models for the calculation (estimation) of some of them is sometimes proposed. Software sensing is thus based on methods that allow calculation of the value of a parameter from the measurement of one or more other parameters, the measurement principle of which is completely different from an existing standard/reference method, or has no direct relation. Statistical correlative methods can also be considered in this group. Some examples will be presented in the following section. 

First principle mathematical models These models solve the basic conservation equations for mass and momentum in their form as... 

Simplified mathematical models These models typically begin with the basic conservation equations of the first principle models but make simplifying assumptions (typically related to similarity theory) to reduce the problem to the solution of (simultaneous) ordinary differential equations. In the verification process, such models must also address the relevant physical phenomenon as well as be validated for the application being considered. Such models are typically easily solved on a computer with typically less user interaction than required for the solution of PDEs. Simplified mathematical models may also be used as screening tools to identify the most important release scenarios however, other modeling approaches should be considered only if they address and have been validated for the important aspects of the scenario under consideration. 

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