Operations Involving Complex Variables

However, these formal expressions are of no great help until we know how to operate with the exponential operator, which is very complicated because it involves the full iV-body problem with the interactions between the particles. In order to circumvent this difficulty, we shall use a resolvant technique 89 we define a resolvant operator (L — 2)-1, function of the complex variable z, and write ... 

Biofuels are produced in a sequence of large batch operations involving bio/chemical reactions, separation and purification steps, followed by formulation with specific additives. The final product must comply with multiple quality specifications despite the variability in raw materials and the complexity of unit operations used in their processing. 

The generalized Dirac function of a complex variable from Eq. (139) belongs to the class of the so-called ultra distributions [2], In the present context, S (z -uk) has the same operational property as the usual Dirac function with a real argument, except that the contour integrals are involved, viz ... 

It should be observed that since the operators T to be discussed include differential operators, it may be practical from the very beginning to assume that the functions F(Z) and G(Z) are analytic functions of all the complex variables involved. 

The development of a successful tablet formulation can be a substantial challenge, because formulation scientists are often confronted with a bewildering array of formulation and process variables that can interact in complex ways. These interactions will primarily be discussed in Volume 2, but to understand these interactions the reader must first have a good understanding of the different unit operations involved in making a tablet, the physicochemical and mechanical properties of the active drug substance, and the causes of drug product instability. 

Mass-Transfer Models Because the mass-transfer coefficient and interfacial area for mass transfer of solute are complex functions of fluid properties and the operational and geometric variables of a stirred-tank extractor or mixer, the approach to design normally involves scale-up of miniplant data. The mass-transfer coefficient and interfacial area are influenced by numerous factors that are difficult to precisely quantify. These include drop coalescence and breakage rates as well as complex flow patterns that exist within the vessel (a function of impeller type, vessel geometry, and power input). Nevertheless, it is instructive to review available mass-transfer coefficient and interfacial area models for the insights they can offer. 

When the reaction operations involve reversible reactions or competing reactions, the split fractions of the species leaving the separators are complex functions of the operating conditions such as the temperatures, pressures, and reflux ratios, and purge streams exist, then iterative calculations are necessary. In these cases, the simulation flowsheets usually contain information recycle loops, that is, cycles for which too few stream variables are known to permit the equations for each unit to be solved independently. For these processes, a solution technique is needed to solve the equations for all of the units in an information recycle loop. 

Tlie discussions of the basic features of filtration given thus far illustrate that the unit operation involves some rather complicated hydrodynamics that depend strongly on the physical properties of both fluid and particles, as well as interaction with a complex porous medium. The process is essentially influenced by two different groups of factors, which can be broadly lumped into macro- and micro-properties. Macrofactors are related to variables such as the area of a filter medium, pressure differences, cake thickness and the viscosity of the liquid phase. Such parameters are readily measured. Micro-factors include the influences of the size and configuration of pores in the cake and filter medium, the thickness of the electrical double layer on the surface of solid particles, and other properties. 

In addition to these technical aspects, economic aspects should also be taken into account whereas the capital cost of a plant designed by scale-up generally increases with respect to the production to the power of approximately 0.7, the capital cost of plants designed by numbering-up is expected to be proportional to their capacity [13, 19]. This implies that numbering-up should be intrinsically more appropriate for variable production systems not exceeding a given maximum production level and for which the overall value depends on the complexity of the system. Since the various unit operations involved in a system require more or less complex microunits... 

Apart from the degree of novelty of a process event, its complexity (e.g., the range of operations to be carried out), the interrelationships of the process variables involved and the required accuracy, will affect performance. Startup and shutdown operations are examples of tasks which, although are not entirely unfamiliar, involve a high degree of complexity. 

Three-phase slurry reactors are commonly used in fine-chemical industries for the catalytic hydrogenation of organic substrates to a variety of products and intermediates (1-2). The most common types of catalysts are precious metals such as Pt and Pd supported on powdered carbon supports (3). The behavior of the gas-liquid-sluny reactors is affected by a complex interplay of multiple variables including the temperature, pressure, stirring rates, feed composition, etc. (1-2,4). Often these types of reactors are operated away from the optimal conditions due to the difficulty in identifying and optimizing the critical variables involved in the process. This not only leads to lost productivity but also increases the cost of down stream processing (purification), and pollution control (undesired by-products). 

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