Analytical geometry mathematical equations

Before describing numerical methods, the technique of using analytical or pseudo-analytical solutions will be discussed. Strictly speaking, a problem has an analytical solution if a mathematical equation can fully describe the phenomena examined. This is usually reserved for simple geometries, with simple conditions and properties. However, by applying specific assumptions and limiting the scope of the problem to be solved, analytical techniques can be applied to more realistic situations. In... 

Mathematics 18.0 Multivariate calculus, differential equations, linear algebra, analytic geometry... 

FFF process can be theoretically described because (a) the flow regime inside the channel of well-defined geometry can be mathematically represented (see Equation 12.1), and (b) the tractability of the various force fields employed in the different techniques allows one to describe the analyte concentration profile (see Equation 12.3). The retention ratio expresses the retardation of an analyte zone caused by its interaction with the field, and it is given by the ratio of the average velocity of the analyte zone and the average velocity of the carrier liquid (n) ... 

This Chapter is concerned with some of the mathematical tools required to describe special properties of curved surfaces. The tools are to be found in differential geometry, analytical function theory, and topology. General references can be foimd at the end of the Chapter. The reader xminterested in the mathematics can skip the equations and their development. The ideas we want to focus on will be clear enough in the text. A particular class of saddle-shaped (hyperbolic) surfaces called minimal surfaces will be treated with special attention since they are relatively straightforward to treat mathematically and do form good approximate representations of actual physical and chemical structures. 

Since, as discussed above, it is impossible to achieve dynamic similarity between laboratory and full scale, the predictive capability of empirical modeling of crystallization is limited. Mathematical modeling also has its shortcomings. Suspension flows in crystallizers are turbulent, two and perhaps even three phase (for boiling crystallizers), the particle size is distributed, and the geometry is complicated with perhaps multiple moving parts (impellers). This is of course beyond the possibility of analytical solution of the equations of motion, so we must turn to computational fluid dynamics (CFD). However, even CFD is not capable of successfully dealing with all of these features. Successful computational models of crystallizers to date are limited to very specific limited problems. 

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