Mathematical operations conversion factors

In measurement sciences, calibration is an operation that establish a relationship between an output quantity, qouU with an input quantity, q m for a measuring system under specified conditions (qin qout) The result of calibration is a model that may have the form of a conversion factor, a mathematical equation, or a graph. By means of this model, then it is possible to estimate q -values from measured q0Ut-values (qout qin) as can be seen in an abstracted form in Fig. 6.1. 

Dimensional analysis, sometimes called the factor label (unit conversion) method, is a method for setting up mathematical problems. Mathematical operations are conducted with the units associated with the numbers, and these units are cancelled until only the unit of the desired answer is left. This results in a setup for the problem. Then the mathematical operations can efficiently be conducted and the final answer calculated and rounded off to the correct number of significant figures. For example, to determine the number of centimeters in 2.3 miles ... 

Compare and contrast the multiplication/division significant figure rule to the significant figure rule applied for addition/subtraction mathematical operations. Explain how density can be used as a conversion factor to convert the volume of an object to the mass of the object, and vice versa. 

This route requires two additional pieces of data the molar mass of the given substance and the mole ratio. The molar mass is determined by using masses from the periodic table. We will follow a procedure much like the one used previously by using the units of the molar mass conversion factor to guide our mathematical operations. Because the known quantity is a mass, the conversion factor will need to be 1 mol divided by molar mass. This conversion factor cancels units of grams and leaves units of moles (see Figure 2.3 below). 

The relationship between the weight concentration of the element to be analysed and the intensity measured from one of its characteristic spectral lines is a complex one. For trace analysis several mathematical models have been developed to correlate fluorescence to the atomic concentration. A series of corrections must be introduced to account for inter-element interactions, preferential excitation, self-absorption and the fluorescence yield (the heavier atoms relax by internal conversion without photon emission). All of these factors require the reference samples to be practically the same structure and atomic composition than the sample under investigation, for all of the elements present. It is mostly because of these reasons that quantitative analysis by X-ray fluorescence is difficult to obtain. When operating upon a solid sample, a perfectly clean surface is important, preferably polished, since the analysis concerns the composition immediately close to the surface. 

The kinetics of the reaction and the properties of the catalyst, especially the thermal stability, will further narrow the range of possible reaction conditions and define a "window" of possible operating parameters. Process optimization, energy efficiency, and safety aspects will then determine at what conditions within the "window" the reactor should operate to give the optimum result. And then mathematical models are used to determine how big the reactor must be to obtain the performance (conversion and pressure drop) determined by the process optimization. Instrumentation is then considered, proper materials of construction are selected, catalyst loading and unloading is considered, possible transport limitations are determined, workshop manufacture is considered, and at last the design of the reactor is completed. The procedure is, of course, iterative since the reactor cost is one of the parameters in the economical optimization, but, as mentioned above, often a factor of minor importance for the overall result. 

One unique but normally undesirable feature of continuous emulsion polymerization carried out in a stirred tank reactor is reactor dynamics. For example, sustained oscillations (limit cycles) in the number of latex particles per unit volume of water, monomer conversion, and concentration of free surfactant have been observed in continuous emulsion polymerization systems operated at isothermal conditions [52-55], as illustrated in Figure 7.4a. Particle nucleation phenomena and gel effect are primarily responsible for the observed reactor instabilities. Several mathematical models that quantitatively predict the reaction kinetics (including the reactor dynamics) involved in continuous emulsion polymerization can be found in references 56-58. Tauer and Muller [59] developed a kinetic model for the emulsion polymerization of vinyl chloride in a continuous stirred tank reactor. The results show that the sustained oscillations depend on the rates of particle growth and coalescence. Furthermore, multiple steady states have been experienced in continuous emulsion polymerization carried out in a stirred tank reactor, and this phenomenon is attributed to the gel effect [60,61]. All these factors inevitably result in severe problems of process control and product quality. 

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